WebGL Programming Guide: Interactive 3D Graphics With Guide

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Praise for
WebGL Programming Guide
“WebGL provides one of the final features for creating applications that deliver ‘the desktop application experience’ in a web browser, and the WebGL Programming Guide leads the
way in creating those applications. Its coverage of all aspects of using WebGL—JavaScript,
OpenGL ES, and fundamental graphics techniques—delivers a thorough education on everything you need to get going. Web-based applications are the wave of the future, and this
book will get you ahead of the curve!”
Dave Shreiner, Coauthor of The OpenGL Programming Guide, Eighth Edition; Series Editor,
OpenGL Library (Addison Wesley)
“HTML5 is evolving the Web into a highly capable application platform supporting beautiful, engaging, and fully interactive applications that run portably across many diverse
systems. WebGL is a vital part of HTML5, as it enables web programmers to access the
full power and functionality of state-of-the-art 3D graphics acceleration. WebGL has been
designed to run securely on any web-capable system and will unleash a new wave of developer innovation in connected 3D web-content, applications, and user interfaces. This book
will enable web developers to fully understand this new wave of web functionality and
leverage the exciting opportunities it creates.”
Neil Trevett, Vice President Mobile Content, NVIDIA; President, The Khronos Group
“With clear explanations supported by beautiful 3D renderings, this book does wonders in
transforming a complex topic into something approachable and appealing. Even without
denying the sophistication of WebGL, it is an accessible resource that beginners should
consider picking up before anything else.”
Evan Burchard, Author, Web Game Developer’s Cookbook (Addison Wesley)
“Both authors have a strong OpenGL background and transfer this knowledge nicely over
to WebGL, resulting in an excellent guide for beginners as well as advanced readers.”
Daniel Haehn, Research Software Developer, Boston Children’s Hospital
“WebGL Programming Guide provides a straightforward and easy-to-follow look at the mechanics of building 3D applications for the Web without relying on bulky libraries or wrappers. A great resource for developers seeking an introduction to 3D development concepts
mixed with cutting-edge web technology.”
Brandon Jones, Software Engineer, Google

“This is more great work from a brilliant researcher. Kouichi Matsuda shows clear and concise steps to bring the novice along the path of understanding WebGL. This is a complex
topic, but he makes it possible for anyone to start using this exciting new web technology.
And he includes basic 3D concepts to lay the foundation for further learning. This will be a
great addition to any web designer’s library.”
Chris Marrin, WebGL Spec. Editor
“WebGL Programming Guide is a great way to go from a WebGL newbie to a WebGL expert.
WebGL, though simple in concept, requires a lot of 3D math knowledge, and WebGL Programming Guide helps you build this knowledge so you’ll be able to understand and apply
it to your programs. Even if you end up using some other WebGL 3D library, the knowledge learned in WebGL Programming Guide will help you understand what those libraries
are doing and therefore allow you to tame them to your application’s specific needs. Heck,
even if you eventually want to program desktop OpenGL and/or DirectX, WebGL Programming Guide is a great start as most 3D books are outdated relative to current 3D technology.
WebGL Programming Guide will give you the foundation for fully understanding modern 3D
graphics.”
Gregg Tavares, An Implementer of WebGL in Chrome

WebGL Programming
Guide

OpenGL Series
from Addison-Wesley

Visit informit.com/opengl for a complete list of available products.

T

he OpenGL graphics system is a software interface to graphics hardware.
(“GL” stands for “Graphics Library”.) It allows you to create interactive programs

that produce color images of moving, three-dimensional objects. With OpenGL,
you can control computer-graphics technology to produce realistic pictures, or
ones that depart from reality in imaginative ways.
The OpenGL Series from Addison-Wesley Professional comprises tutorial and
reference books that help programmers gain a practical understanding of OpenGL
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WebGL
Programming
Guide:
Interactive 3D
Graphics Programming
with WebGL
Kouichi Matsuda
Rodger Lea

Upper Saddle River, NJ • Boston • Indianapolis • San Francisco
New York • Toronto • Montreal • London • Munich • Paris • Madrid
Cape Town • Sydney • Tokyo • Singapore • Mexico City

Many of the designations used by manufacturers and sellers to distinguish their
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been printed with initial capital letters or in all capitals. OpenGL is a registered
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Khronos and WebGL are trademarks of the Khronos Group Inc. Google, Google
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is a registered trademark of the Mozilla Foundation. Apple, iPhone, Macintosh,
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is a registered trademark of Microsoft Corporation. Nvidia and Nvidia Geforce
are trademarks of NVIDIA Corporation. AMD and Radeon are trademarks of
Advanced Micro Devices, Inc.

Editor-in-Chief
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programs contained herein.

Senior Indexer
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Thoughts are filled along with time, the distant days will not return,
and time passed is like a spiral of semiprecious stones...
—Kouichi Matsuda

To my wife, family, and friends—for making life fun.
—Rodger Lea

Contents
Preface
1.

Overview of WebGL

xvii
1

Advantages of WebGL ....................................................................................................3
You Can Start Developing 3D Graphics Applications Using Only a Text Editor .....3
Publishing Your 3D Graphics Applications Is Easy ...................................................4
You Can Leverage the Full Functionality of the Browser .........................................5
Learning and Using WebGL Is Easy...........................................................................5
Origins of WebGL ...........................................................................................................5
Structure of WebGL Applications ...................................................................................6
Summary ........................................................................................................................7
2.

Your First Step with WebGL

9

What Is a Canvas? ..........................................................................................................9
Using the  Tag ...........................................................................................11
DrawRectangle.js ......................................................................................................13
The World’s Shortest WebGL Program: Clear Drawing Area ......................................16
The HTML File (HelloCanvas.html).........................................................................17
JavaScript Program (HelloCanvas.js)........................................................................18
Experimenting with the Sample Program ...............................................................23
Draw a Point (Version 1) ..............................................................................................23
HelloPoint1.html......................................................................................................25
HelloPoint1.js ...........................................................................................................25
What Is a Shader?.....................................................................................................27
The Structure of a WebGL Program that Uses Shaders ...........................................28
Initializing Shaders ...................................................................................................30
Vertex Shader ...........................................................................................................33
Fragment Shader ......................................................................................................35
The Draw Operation ................................................................................................36
The WebGL Coordinate System...............................................................................38
Experimenting with the Sample Program ...............................................................40
Draw a Point (Version 2) ..............................................................................................41
Using Attribute Variables .........................................................................................41
Sample Program (HelloPoint2.js) .............................................................................42
Getting the Storage Location of an Attribute Variable ...........................................44
Assigning a Value to an Attribute Variable .............................................................45
Family Methods of gl.vertexAttrib3f() .....................................................................47
Experimenting with the Sample Program ...............................................................49

Draw a Point with a Mouse Click.................................................................................50
Sample Program (ClickedPoints.js) ..........................................................................50
Register Event Handlers ...........................................................................................52
Handling Mouse Click Events..................................................................................53
Experimenting with the Sample Program ...............................................................57
Change the Point Color................................................................................................58
Sample Program (ColoredPoints.js) .........................................................................59
Uniform Variables ....................................................................................................61
Retrieving the Storage Location of a Uniform Variable ..........................................62
Assigning a Value to a Uniform Variable ................................................................63
Family Methods of gl.uniform4f() ...........................................................................65
Summary ......................................................................................................................66
3.

Drawing and Transforming Triangles

67

Drawing Multiple Points...............................................................................................68
Sample Program (MultiPoint.js) ...............................................................................70
Using Buffer Objects ................................................................................................72
Create a Buffer Object (gl.createBuffer()) .................................................................74
Bind a Buffer Object to a Target (gl.bindBuffer()) ...................................................75
Write Data into a Buffer Object (gl.bufferData())....................................................76
Typed Arrays.............................................................................................................78
Assign the Buffer Object to an Attribute Variable
(gl.vertexAttribPointer()) ..........................................................................................79
Enable the Assignment to an Attribute Variable (gl.enableVertexAttribArray()) ...81
The Second and Third Parameters of gl.drawArrays() .............................................82
Experimenting with the Sample Program ...............................................................84
Hello Triangle ...............................................................................................................85
Sample Program (HelloTriangle.js) ..........................................................................85
Basic Shapes ..............................................................................................................87
Experimenting with the Sample Program ...............................................................89
Hello Rectangle (HelloQuad) ...................................................................................89
Experimenting with the Sample Program ...............................................................91
Moving, Rotating, and Scaling .....................................................................................91
Translation ...............................................................................................................92
Sample Program (TranslatedTriangle.js) ..................................................................93
Rotation ....................................................................................................................96
Sample Program (RotatedTriangle.js).......................................................................99
Transformation Matrix: Rotation ..........................................................................102
Transformation Matrix: Translation ......................................................................105
Rotation Matrix, Again ..........................................................................................106
Sample Program (RotatedTriangle_Matrix.js) ........................................................107

Contents

ix

Reusing the Same Approach for Translation .........................................................111
Transformation Matrix: Scaling .............................................................................111
Summary ....................................................................................................................113
4.

More Transformations and Basic Animation

115

Translate and Then Rotate .........................................................................................115
Transformation Matrix Library: cuon-matrix.js ....................................................116
Sample Program (RotatedTriangle_Matrix4.js) ......................................................117
Combining Multiple Transformation ....................................................................119
Sample Program (RotatedTranslatedTriangle.js) ....................................................121
Experimenting with the Sample Program .............................................................123
Animation ...................................................................................................................124
The Basics of Animation ........................................................................................125
Sample Program (RotatingTriangle.js) ...................................................................126
Repeatedly Call the Drawing Function (tick())......................................................129
Draw a Triangle with the Specified Rotation Angle (draw()) ................................130
Request to Be Called Again (requestAnimationFrame()) .......................................131
Update the Rotation Angle (animate()) .................................................................133
Experimenting with the Sample Program .............................................................135
Summary ....................................................................................................................136
5.

Using Colors and Texture Images

137

Passing Other Types of Information to Vertex Shaders .............................................137
Sample Program (MultiAttributeSize.js).................................................................139
Create Multiple Buffer Objects ..............................................................................140
The gl.vertexAttribPointer() Stride and Offset Parameters ....................................141
Sample Program (MultiAttributeSize_Interleaved.js) .............................................142
Modifying the Color (Varying Variable)................................................................146
Sample Program (MultiAttributeColor.js) ..............................................................147
Experimenting with the Sample Program .............................................................150
Color Triangle (ColoredTriangle.js) ............................................................................151
Geometric Shape Assembly and Rasterization ......................................................151
Fragment Shader Invocations ................................................................................155
Experimenting with the Sample Program .............................................................156
Functionality of Varying Variables and the Interpolation Process .......................157
Pasting an Image onto a Rectangle ............................................................................160
Texture Coordinates ...............................................................................................162
Pasting Texture Images onto the Geometric Shape ..............................................162
Sample Program (TexturedQuad.js) .......................................................................163
Using Texture Coordinates (initVertexBuffers()) ...................................................166
Setting Up and Loading Images (initTextures()) ...................................................166
Make the Texture Ready to Use in the WebGL System (loadTexture()) ...............170
x

WebGL Programming Guide

Flip an Image’s Y-Axis ............................................................................................170
Making a Texture Unit Active (gl.activeTexture()) ................................................171
Binding a Texture Object to a Target (gl.bindTexture()).......................................173
Set the Texture Parameters of a Texture Object (gl.texParameteri()) ....................174
Assigning a Texture Image to a Texture Object (gl.texImage2D()) .......................177
Pass the Texture Unit to the Fragment Shader (gl.uniform1i()) ...........................179
Passing Texture Coordinates from the Vertex Shader to the Fragment Shader ...180
Retrieve the Texel Color in a Fragment Shader (texture2D()) ..............................181
Experimenting with the Sample Program .............................................................182
Pasting Multiple Textures to a Shape .........................................................................183
Sample Program (MultiTexture.js) .........................................................................184
Summary ....................................................................................................................189
6.

The OpenGL ES Shading Language (GLSL ES)

191

Recap of Basic Shader Programs .................................................................................191
Overview of GLSL ES ..................................................................................................192
Hello Shader! ...............................................................................................................193
Basics ......................................................................................................................193
Order of Execution .................................................................................................193
Comments ..............................................................................................................193
Data (Numerical and Boolean Values) .......................................................................194
Variables ....................................................................................................................194
GLSL ES Is a Type Sensitive Language ........................................................................195
Basic Types ..................................................................................................................195
Assignment and Type Conversion .........................................................................196
Operations ..............................................................................................................197
Vector Types and Matrix Types ..................................................................................198
Assignments and Constructors ..............................................................................199
Access to Components ...........................................................................................201
Operations ..............................................................................................................204
Structures ....................................................................................................................207
Assignments and Constructors ..............................................................................207
Access to Members .................................................................................................207
Operations ..............................................................................................................208
Arrays
....................................................................................................................208
Samplers ....................................................................................................................209
Precedence of Operators .............................................................................................210
Conditional Control Flow and Iteration ....................................................................211
if Statement and if-else Statement .........................................................................211
for Statement ..........................................................................................................211
continue, break, discard Statements ......................................................................212

Contents

xi

Functions ....................................................................................................................213
Prototype Declarations ...........................................................................................214
Parameter Qualifiers ...............................................................................................214
Built-In Functions .......................................................................................................215
Global Variables and Local Variables .........................................................................216
Storage Qualifiers ........................................................................................................217
const Variables .......................................................................................................217
Attribute Variables..................................................................................................218
Uniform Variables ..................................................................................................218
Varying Variables ...................................................................................................219
Precision Qualifiers .....................................................................................................219
Preprocessor Directives ...............................................................................................221
Summary ....................................................................................................................223
7.

Toward the 3D World

225

What’s Good for Triangles Is Good for Cubes ...........................................................225
Specifying the Viewing Direction...............................................................................226
Eye Point, Look-At Point, and Up Direction .........................................................227
Sample Program (LookAtTriangles.js) ....................................................................229
Comparing LookAtTriangles.js with RotatedTriangle_Matrix4.js .........................232
Looking at Rotated Triangles from a Specified Position .......................................234
Sample Program (LookAtRotatedTriangles.js) .......................................................235
Experimenting with the Sample Program .............................................................236
Changing the Eye Point Using the Keyboard........................................................238
Sample Program (LookAtTrianglesWithKeys.js) ....................................................238
Missing Parts ..........................................................................................................241
Specifying the Visible Range (Box Type) ....................................................................241
Specify the Viewing Volume ..................................................................................242
Defining a Box-Shaped Viewing Volume ..............................................................243
Sample Program (OrthoView.html) .......................................................................245
Sample Program (OrthoView.js) ............................................................................246
Modifying an HTML Element Using JavaScript ....................................................247
The Processing Flow of the Vertex Shader ............................................................248
Changing Near or Far.............................................................................................250
Restoring the Clipped Parts of the Triangles
(LookAtTrianglesWithKeys_ViewVolume.js) .........................................................251
Experimenting with the Sample Program .............................................................253
Specifying the Visible Range Using a Quadrangular Pyramid ...................................254
Setting the Quadrangular Pyramid Viewing Volume ............................................256
Sample Program (PerspectiveView.js) ....................................................................258
The Role of the Projection Matrix .........................................................................260
Using All the Matrices (Model Matrix, View Matrix, and Projection Matrix) ......262
xii

WebGL Programming Guide

Sample Program (PerspectiveView_mvp.js) ...........................................................263
Experimenting with the Sample Program .............................................................266
Correctly Handling Foreground and Background Objects ........................................267
Hidden Surface Removal ........................................................................................270
Sample Program (DepthBuffer.js) ..........................................................................272
Z Fighting ...............................................................................................................273
Hello Cube ..................................................................................................................275
Drawing the Object with Indices and Vertices Coordinates .................................277
Sample Program (HelloCube.js) .............................................................................278
Writing Vertex Coordinates, Colors, and Indices to the Buffer Object ................281
Adding Color to Each Face of a Cube....................................................................284
Sample Program (ColoredCube.js) .........................................................................285
Experimenting with the Sample Program .............................................................287
Summary ....................................................................................................................289
8.

Lighting Objects

291

Lighting 3D Objects ....................................................................................................291
Types of Light Source .............................................................................................293
Types of Reflected Light .........................................................................................294
Shading Due to Directional Light and Its Diffuse Reflection ...............................296
Calculating Diffuse Reflection Using the Light Direction and the
Orientation of a Surface .........................................................................................297
The Orientation of a Surface: What Is the Normal? .............................................299
Sample Program (LightedCube.js) .........................................................................302
Add Shading Due to Ambient Light ......................................................................307
Sample Program (LightedCube_ambient.js) ..........................................................308
Lighting the Translated-Rotated Object .....................................................................310
The Magic Matrix: Inverse Transpose Matrix ........................................................311
Sample Program (LightedTranslatedRotatedCube.js) ............................................312
Using a Point Light Object .........................................................................................314
Sample Program (PointLightedCube.js) .................................................................315
More Realistic Shading: Calculating the Color per Fragment...............................319
Sample Program (PointLightedCube_perFragment.js) ..........................................319
Summary ....................................................................................................................321
9.

Hierarchical Objects

323

Drawing and Manipulating Objects Composed of Other Objects ............................324
Hierarchical Structure.............................................................................................325
Single Joint Model ..................................................................................................326
Sample Program (JointModel.js) ............................................................................328
Draw the Hierarchical Structure (draw()) ..............................................................332
A Multijoint Model ................................................................................................334
Contents

xiii

Sample Program (MultiJointModel.js) ...................................................................335
Draw Segments (drawBox())...................................................................................339
Draw Segments (drawSegment()) ...........................................................................340
Shader and Program Objects: The Role of initShaders() ............................................344
Create Shader Objects (gl.createShader()) ..............................................................345
Store the Shader Source Code in the Shader Objects (g.shaderSource())..............346
Compile Shader Objects (gl.compileShader()) .......................................................347
Create a Program Object (gl.createProgram()) .......................................................349
Attach the Shader Objects to the Program Object (gl.attachShader()) .................350
Link the Program Object (gl.linkProgram())..........................................................351
Tell the WebGL System Which Program Object to Use (gl.useProgram())...........353
The Program Flow of initShaders() ........................................................................353
Summary ....................................................................................................................356
10. Advanced Techniques

357

Rotate an Object with the Mouse...............................................................................357
How to Implement Object Rotation ......................................................................358
Sample Program (RotateObject.js) .........................................................................358
Select an Object ..........................................................................................................360
How to Implement Object Selection .....................................................................361
Sample Program (PickObject.js) .............................................................................362
Select the Face of the Object..................................................................................365
Sample Program (PickFace.js).................................................................................366
HUD (Head Up Display) .............................................................................................368
How to Implement a HUD.....................................................................................369
Sample Program (HUD.html) .................................................................................369
Sample Program (HUD.js) ......................................................................................370
Display a 3D Object on a Web Page (3DoverWeb) ...............................................372
Fog (Atmospheric Effect) ............................................................................................372
How to Implement Fog ..........................................................................................373
Sample Program (Fog.js).........................................................................................374
Use the w Value (Fog_w.js) ....................................................................................376
Make a Rounded Point ...............................................................................................377
How to Implement a Rounded Point ....................................................................377
Sample Program (RoundedPoints.js)......................................................................378
Alpha Blending ...........................................................................................................380
How to Implement Alpha Blending ......................................................................380
Sample Program (LookAtBlendedTriangles.js) .......................................................381
Blending Function..................................................................................................382
Alpha Blend 3D Objects (BlendedCube.js) ............................................................384
How to Draw When Alpha Values Coexist ...........................................................385

xiv

WebGL Programming Guide

Switching Shaders .......................................................................................................386
How to Implement Switching Shaders ..................................................................387
Sample Program (ProgramObject.js) ......................................................................387
Use What You’ve Drawn as a Texture Image .............................................................392
Framebuffer Object and Renderbuffer Object .......................................................392
How to Implement Using a Drawn Object as a Texture .......................................394
Sample Program (FramebufferObjectj.js) ...............................................................395
Create Frame Buffer Object (gl.createFramebuffer()) .............................................397
Create Texture Object and Set Its Size and Parameters .........................................397
Create Renderbuffer Object (gl.createRenderbuffer()) ...........................................398
Bind Renderbuffer Object to Target and Set Size (gl.bindRenderbuffer(),
gl.renderbufferStorage()) ........................................................................................399
Set Texture Object to Framebuffer Object (gl.bindFramebuffer(),
gl.framebufferTexture2D()) ....................................................................................400
Set Renderbuffer Object to Framebuffer Object
(gl.framebufferRenderbuffer()) ...............................................................................401
Check Configuration of Framebuffer Object (gl.checkFramebufferStatus()) ........402
Draw Using the Framebuffer Object ......................................................................403
Display Shadows .........................................................................................................405
How to Implement Shadows .................................................................................405
Sample Program (Shadow.js)..................................................................................406
Increasing Precision ...............................................................................................412
Sample Program (Shadow_highp.js) ......................................................................413
Load and Display 3D Models .....................................................................................414
The OBJ File Format ...............................................................................................417
The MTL File Format ..............................................................................................418
Sample Program (OBJViewer.js) .............................................................................419
User-Defined Object ...............................................................................................422
Sample Program (Parser Code in OBJViewer.js) ....................................................423
Handling Lost Context ...............................................................................................430
How to Implement Handling Lost Context ..........................................................431
Sample Program (RotatingTriangle_contextLost.js) ..............................................432
Summary ....................................................................................................................434
A.

No Need to Swap Buffers in WebGL

437

B.

Built-in Functions of GLSL ES 1.0

441

Angle and Trigonometry Functions ...........................................................................441
Exponential Functions ................................................................................................443
Common Functions ....................................................................................................444
Geometric Functions ..................................................................................................447

Contents

xv

Matrix Functions.........................................................................................................448
Vector Functions .........................................................................................................449
Texture Lookup Functions ..........................................................................................451
C.

Projection Matrices

453

Orthogonal Projection Matrix ....................................................................................453
Perspective Projection Matrix .....................................................................................453
D.

WebGL/OpenGL: Left or Right Handed?

455

Sample Program CoordinateSystem.js ........................................................................456
Hidden Surface Removal and the Clip Coordinate System .......................................459
The Clip Coordinate System and the Viewing Volume.............................................460
What Is Correct? .........................................................................................................462
Summary ....................................................................................................................464
E.

The Inverse Transpose Matrix

465

F.

Load Shader Programs from Files

471

G.

World Coordinate System Versus Local Coordinate System

473

The Local Coordinate System .....................................................................................474
The World Coordinate System ...................................................................................475
Transformations and the Coordinate Systems ...........................................................477
H.

xvi

Web Browser Settings for WebGL

479

Glossary

481

References

485

Index

487

WebGL Programming Guide

Preface
WebGL is a technology that enables drawing, displaying, and interacting with sophisticated interactive three-dimensional computer graphics (“3D graphics”) from within
web browsers. Traditionally, 3D graphics has been restricted to high-end computers or
dedicated game consoles and required complex programming. However, as both personal
computers and, more importantly, web browsers have become more sophisticated, it has
become possible to create and display 3D graphics using accessible and well-known web
technologies. This book provides a comprehensive overview of WebGL and takes the
reader, step by step, through the basics of creating WebGL applications. Unlike other
3D graphics technologies such as OpenGL and Direct3D, WebGL applications can be
constructed as web pages so they can be directly executed in the browsers without installing any special plug-ins or libraries. Therefore, you can quickly develop and try out a
sample program with a standard PC environment; because everything is web based, you
can easily publish the programs you have constructed on the web. One of the promises
of WebGL is that, because WebGL applications are constructed as web pages, the same
program can be run across a range of devices, such as smart phones, tablets, and game
consoles, through the browser. This powerful model means that WebGL will have a significant impact on the developer community and will become one of the preferred tools for
graphics programming.

Who the Book Is For
We had two main audiences in mind when we wrote this book: web developers looking
to add 3D graphics to their web pages and applications, and 3D graphics programmers
wishing to understand how to apply their knowledge to the web environment. For web
developers who are familiar with standard web technologies such as HTML and JavaScript
and who are looking to incorporate 3D graphics into their web pages or web applications, WebGL offers a simple yet powerful solution. It can be used to add 3D graphics to
enhance web pages, to improve the user interface (UI) for a web application by using a 3D
interface, and even to develop more complex 3D applications and games that run in web
browsers.
The second target audience is programmers who have worked with one of the main 3D
application programming interfaces (APIs), such as Direct3D or OpenGL, and who are
interested in understanding how to apply their knowledge to the web environment. We
would expect these programmers to be interested in the more complex 3D applications
that can be developed in modern web browsers.
However, the book has been designed to be accessible to a wide audience using a step-bystep approach to introduce features of WebGL, and it assumes no background in 2D or 3D
graphics. As such, we expect it also to be of interest to the following:

Preface

xvii

• General programmers seeking an understanding of how web technologies are evolving in the graphics area

• Students studying 2D and 3D graphics because it offers a simple way to begin to experiment with graphics via a web browser rather than setting up a full programming
environment

• Web developers exploring the “bleeding edge” of what is possible on mobile devices
such as Android or iPhone using the latest mobile web browsers

What the Book Covers
This book covers the WebGL 1.0 API along with all related JavaScript functions. You will
learn how HTML, JavaScript, and WebGL are related, how to set up and run WebGL applications, and how to incorporate sophisticated 3D program “shaders” under the control
of JavaScript. The book details how to write vertex and fragment shaders, how to implement advanced rendering techniques such as per-pixel lighting and shadowing, and basic
interaction techniques such as selecting 3D objects. Each chapter develops a number of
working, fully functional WebGL applications and explains key WebGL features through
these examples. After finishing the book, you will be ready to write WebGL applications
that fully harness the programmable power of web browsers and the underlying graphics
hardware.

How the Book Is Structured
This book is organized to cover the API and related web APIs in a step-by-step fashion,
building up your knowledge of WebGL as you go.

Chapter 1—Overview of WebGL
This chapter briefly introduces you to WebGL, outlines some of the key features and
advantages of WebGL, and discusses its origins. It finishes by explaining the relationship
of WebGL to HTML5 and JavaScript and which web browsers you can use to get started
with your exploration of WebGL.

Chapter 2—Your First Step with WebGL
This chapter explains the  element and the core functions of WebGL by taking
you, step-by-step, through the construction of several example programs. Each example
is written in JavaScript and uses WebGL to display and interact with a simple shape on
a web page. The example WebGL programs will highlight some key points, including:
(1) how WebGL uses the  element object and how to draw on it; (2) the linkage
between HTML and WebGL using JavaScript; (3) simple WebGL drawing functions; and
(4) the role of shader programs within WebGL.

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WebGL Programming Guide

Chapter 3—Drawing and Transforming Triangles
This chapter builds on those basics by exploring how to draw more complex shapes and
how to manipulate those shapes in 3D space. This chapter looks at: (1) the critical role of
triangles in 3D graphics and WebGL’s support for drawing triangles; (2) using multiple
triangles to draw other basic shapes; (3) basic transformations that move, rotate, and
scale triangles using simple equations; and (4) how matrix operations make transformations simple.

Chapter 4—More Transformations and Basic Animation
In this chapter, you explore further transformations and begin to combine transformations
into animations. You: (1) are introduced to a matrix transformation library that hides the
mathematical details of matrix operations; (2) use the library to quickly and easily combine
multiple transformations; and (3) explore animation and how the library helps you
animate simple shapes. These techniques provide the basics to construct quite complex
WebGL programs and will be used in the sample programs in the following chapters.

Chapter 5—Using Colors and Texture Images
Building on the basics described in previous chapters, you now delve a little further into
WebGL by exploring the following three subjects: (1) besides passing vertex coordinates,
how to pass other data such as color information to the vertex shader; (2) the conversion from a shape to fragments that takes place between the vertex shader and the fragment shader, which is known as the rasterization process; and (3) how to map images (or
textures) onto the surfaces of a shape or object. This chapter is the final chapter focusing
on the key functionalities of WebGL.

Chapter 6—The OpenGL ES Shading Language (GLSL ES)
This chapter takes a break from examining WebGL sample programs and explains the core
features of the OpenGL ES Shading Language (GLSL ES) in detail. You will cover: (1) data,
variables, and variable types; (2) vector, matrix, structure, array, and sampler; (3) operators, control flow, and functions; (4) attributes, uniforms, and varyings; (5) precision
qualifier; and (6) preprocessor and directives. By the end of this chapter you will have a
good understanding of GLSL ES and how it can be used to write a variety of shaders.

Chapter 7—Toward the 3D World
This chapter takes the first step into the 3D world and explores the implications of
moving from 2D to 3D. In particular, you will explore: (1) representing the user’s view
into the 3D world; (2) how to control the volume of 3D space that is viewed; (3) clipping;
(4) foreground and background objects; and (5) drawing a 3D object—a cube. All these
issues have a significant impact on how the 3D scene is drawn and presented to viewers.
A mastery of them is critical to building compelling 3D scenes.

Preface

xix

Chapter 8—Lighting Objects
This chapter focuses on lighting objects, looking at different light sources and their effects
on the 3D scene. Lighting is essential if you want to create realistic 3D scenes because it
helps to give the scene a sense of depth.
The following key points are discussed in this chapter: (1) shading, shadows, and different
types of light sources including point, directional, and ambient; (2) reflection of light in
the 3D scene and the two main types: diffuse and ambient reflection; and (3) the details of
shading and how to implement the effect of light to make objects look three-dimensional.

Chapter 9—Hierarchical Objects
This chapter is the final chapter describing the core features and how to program with
WebGL. Once completed, you will have mastered the basics of WebGL and will have
enough knowledge to be able to create realistic and interactive 3D scenes. This chapter
focuses on hierarchical objects, which are important because they allow you to progress
beyond single objects like cubes or blocks to more complex objects that you can use for
game characters, robots, and even modeling humans.

Chapter 10—Advanced Techniques
This chapter touches on a variety of important techniques that use what you have learned
so far and provide you with an essential toolkit for building interactive, compelling 3D
graphics. Each technique is introduced through a complete example, which you can reuse
when building your own WebGL applications.

Appendix A—No Need to Swap Buffers in WebGL
This appendix explains why WebGL programs don’t need to swap buffers.

Appendix B—Built-In Functions of GLSL ES 1.0
This appendix provides a reference for all the built-in functions available in the OpenGL
ES Shading Language.

Appendix C—Projection Matrices
This appendix provides the projection matrices generated by Matrix4.setOrtho() and
Matrix4.setPerspective().

Appendix D—WebGL/OpenGL: Left or Right Handed?
This appendix explains how WebGL and OpenGL deal internally with the coordinate system and clarify that technically, both WebGL and OpenGL are agnostic as to
handedness.
xx

WebGL Programming Guide

Appendix E—The Inverse Transpose Matrix
This appendix explains how the inverse transpose matrix of the model matrix can deal
with the transformation of normal vectors.

Appendix F—Loading Shader Programs from Files
This appendix explains how to load the shader programs from files.

Appendix G—World Coordinate System Versus Local Coordinate System
This appendix explains the different coordinate systems and how they are used in 3D
graphics.

Appendix H—Web Browser Settings for WebGL
This appendix explains how to use advanced web browser settings to ensure that WebGL
is displayed correctly, and what to do if it isn’t.

WebGL-Enabled Browsers
At the time of writing, WebGL is supported by Chrome, Firefox, Safari, and Opera. Sadly,
some browsers, such as IE9 (Microsoft Internet Explorer), don’t yet support WebGL. In
this book, we use the Chrome browser released by Google, which, in addition to WebGL
supports a number of useful features such as a console function for debugging. We have
checked the sample programs in this book using the following environment (Table P.1)
but would expect them to work with any browser supporting WebGL.
Table P.1

PC Environment

Browser

Chrome (25.0.1364.152 m)

OS

Windows 7 and 8

Graphics boards

NVIDIA Quadro FX 380, NVIDIA GT X 580, NVIDIA GeForce GTS 450,
Mobile Intel 4 Series Express Chipset Family, AMD Radeon HD
6970

Refer to the www.khronos.org/webgl/wiki/BlacklistsAndWhitelists for an updated list of
which hardware cards are known to cause problems.
To confirm that you are up and running, download Chrome (or use your preferred
browser) and point it to the companion website for this book at https://sites.google.com/
site/webglbook/
Navigate to Chapter 3 and click the link to the sample file HelloTriangle.html. If you can
see a red triangle as shown in Figure P.1 in the browser, WebGL is working.

Preface

xxi

Figure P.1

Loading HelloTriangle results in a red triangle

If you don’t see the red triangle shown in the figure, take a look at Appendix H, which
explains how to change your browser settings to load WebGL.

Sample Programs and Related Links
All sample programs in this book and related links are available on the companion websites. The official site hosted by the publisher is www.informit.com/
title/9780321902924 and the author site is hosted at https://sites.google.com/site/
webglbook/.
The latter site contains the links to each sample program in this book. You can run each
one directly by clicking the links.
If you want to modify the sample programs, you can download the zip file of all the
samples, available on both sites, to your local disk. In this case, you should note that
the sample program consists of both the HTML file and the associated JavaScript file
in the same folder. For example, for the sample program HelloTriangle, you need
both HelloTriangle.html and HelloTriangle.js. To run HelloTriangle, double-click
HelloTriangle.html.

xxii

WebGL Programming Guide

Style Conventions
These style conventions are used in this book:

• Bold—First occurrences of key terms and important words
• Italic—Parameter names and names of references
•

Monospace—Code examples, methods, functions, variables, command options,
JavaScript object names, filenames, and HTML tags

Preface

xxiii

Acknowledgments
We have been fortunate to receive help and guidance from many talented individuals
during the process of creating this book, both with the initial Japanese version and the
subsequent English one.
Takafumi Kanda helped by providing numerous code samples for our support libraries and sample programs; without him, this book could not have been realized. Yasuko
Kikuchi, Chie Onuma, and Yuichi Nishizawa provided valuable feedback on early versions
of the book. Of particular note, one insightful comment by Ms. Kikuchi literally stopped
the writing, causing a reevaluation of several sections and leading to a much stronger
book. Hiroyuki Tanaka and Kazsuhira Oonishi (iLinx) gave excellent support with the
sample programs, and Teruhisa Kamachi and Tetsuo Yoshitani supported the writing of
sections on HTML5 and JavaScript. The WebGL working group, especially Ken Russell
(Google), Chris Marin (Apple), and Dan Ginsburg (AMD), have answered many technical questions. We have been privileged to receive an endorsement from the president
of the Khronos Group, Neil Trevett, and appreciate the help of Hitoshi Kasai (Principal,
MIACIS Associates) who provided the connection to Mr. Trevett and the WebGL working
group. In addition, thank you to Xavier Michel and Makoto Sato (Sophia University), who
greatly helped with the translation of the original text and issues that arose during the
translation. For the English version, Jeff Gilbert, Rick Rafey, and Daniel Haehn reviewed
this book carefully and gave us excellent technical comments and feedback that greatly
improved the book. Our thanks also to Laura Lewin and Olivia Basegio from Pearson, who
have helped with organizing the publication and ensuring the whole process has been as
smooth and as painless as possible.
We both owe a debt of gratitude to the authors of the “Red Book” (OpenGL Programming
Guide) and the “Gold Book” (OpenGL ES 2.0 Programming Guide) both published by
Pearson, without which this book would not have been possible. We hope, in some small
way, that this book repays some of that debt.

xxiv

WebGL Programming Guide

About the Authors
Dr. Kouichi Matsuda has a broad background in user interface and user experience design
and its application to novel multimedia products. His work has taken him from product
development, through research, and back to development, having spent time at NEC,
Sony Corporate Research, and Sony Computer Science Laboratories. He is currently a chief
distinguished researcher focused on user experience and human computer interaction
across a range of consumer electronics. He was the designer of the social 3D virtual world
called “PAW” (personal agent-oriented virtual world), was involved in the development
of the VRML97 (ISO/IEC 14772-1:1997) standard from the start, and has remained active
in both VRML and X3D communities (precursors to WebGL). He has written 15 books on
computer technologies and translated a further 25 into Japanese. His expertise covers user
experiences, user interface, human computer interaction, natural language understanding,
entertainment-oriented network services, and interface agent systems. Always on
the lookout for new and exciting possibilities in the technology space, he combines his
professional life with a love of hot springs, sea in summer, wines, and MANGA (at
which he dabbles in drawing and illustrations). He received his Ph.D. (Engineering)
from the Graduate School of Engineering, University of Tokyo, and can be reached via
WebGL.prog.guide@gmail.com.
Dr. Rodger Lea is an adjunct professor with the Media and Graphics Interdisciplinary
Centre at the University of British Columbia, with an interest in systems aspects of multimedia and distributed computing. With more than 20 years of experience leading research
groups in both academic and industrial settings, he has worked on early versions of shared
3D worlds, helped define VRML97, developed multimedia operating systems, prototyped
interactive digital TV, and led developments on multimedia home networking standards.
He has published more than 60 research papers and three books, and he holds 12 patents.
His current research explores the growing “Internet of Things,” but he retains a passion
for all things media and graphics.

About the Authors

xxv

This page intentionally left blank

Chapter 1

Overview of WebGL

WebGL is a technology that enables drawing, displaying, and interacting with sophisticated
interactive three-dimensional computer graphics (“3D graphics”) from within web browsers.
Traditionally, 3D graphics has been restricted to high-end computers or dedicated game consoles
and has required complex programming. However, as both personal computers and, more importantly, web browsers, have become more sophisticated, it has become possible to create and
display 3D graphics using accessible and well-known web technologies. WebGL, when combined
with HTML5 and JavaScript, makes 3D graphics accessible to web developers and will play an
important role in the development of next generation, easy-to-use and intuitive user interfaces
and web content. Some examples of this are shown in Figure 1.1. Over the next few years, you
can expect to see WebGL used on a range of devices from standard PCs to consumer electronics,
smart phones, and tablets.

Figure 1.1 Complex 3D graphics within a browser. © 2011 Hiromasa Horie (left), 2012
Kouichi Matsuda (right)
HTML5, the latest evolution of the HTML standard, expands traditional HTML with
features covering 2D graphics, networking, and local storage access. With the advent of
HTML5, browsers are rapidly evolving from simple presentation engines to sophisticated
application platforms. With this evolution comes a need for interface and graphics capabilities beyond 2D. WebGL has been designed for that central role of creating the visual
layer for new browser-based 3D applications and experiences.
Traditionally, creating compelling 3D graphics required you to create a stand-alone application using a programming language such as C or C++ along with dedicated computer
graphics libraries such as OpenGL and Direct3D. However, with WebGL, you can now
realize 3D graphics as part of a standard web page using familiar HTML and JavaScript—
with a little extra code for the 3D graphics.
Importantly, because WebGL is supported as the browser’s default built-in technology for
rendering 3D graphics, you can use WebGL directly without having to install special plugins or libraries. Better still, because it’s all browser based, you can run the same WebGL
applications on various platforms, from sophisticated PCs down to consumer electronics,
tablets, and smart phones.
This chapter briefly introduces you to WebGL, outlines some of the key features and
advantages of WebGL, and discusses its origins. It also explains the relationship of WebGL
to HTML5 and JavaScript and the structure of WebGL programs.

2

CHAPTER 1

Overview of WebGL

Advantages of WebGL
As HTML has evolved, web developers have been able to create increasingly sophisticated
web-based applications. Originally, HTML offered only static content, but the introduction of scripting support like JavaScript enabled more complex interactions and dynamic
content. HTML5 introduced further sophistication, including support for 2D graphics via
the canvas tag. This allowed a variety of graphical elements on a web page, ranging from
dancing cartoon characters to map animations that respond to user input by updating the
maps in real time.
WebGL takes this one step further, enabling the display and manipulation of 3D graphics
on web pages by using JavaScript. Using WebGL, it becomes possible to create rich user
interfaces and 3D games and to use 3D to visualize and manipulate a variety of information from the Internet. Although the technical capabilities of WebGL are impressive, it is
perhaps the ease of use and accessibility that differentiate it from other technologies and
that will ensure its impact. In particular:

• You can start developing 3D graphics applications using only a text editor and
browser.

• You can easily publish the 3D graphics applications using standard web technologies, making them available to your friends or other web users.

• You can leverage the full functionality of the browser.
• Learning and using WebGL is easy because a lot of material is already available for
study and development.

You Can Start Developing 3D Graphics Applications Using Only
a Text Editor
One handy and convenient point in developing applications using WebGL is that you
don’t need to set up an application developing environment for WebGL. As explained
earlier, because WebGL is built into the browser, there is no need for special application development tools such as compilers and linkers to create 3D graphics applications.
As a minimum, to view the sample programs explained in this book, you only need a
WebGL-enabled browser. If you want to edit them or create your own, a standard text
editor (for example, Notepad or TextEdit) is enough. In Figure 1.2, you can see a WebGL
application running in Chrome and the HTML file opened in Notepad. The JavaScript file
(RotateObject.js) that uses WebGL is loaded by the HTML file and could also be edited
using a simple text editor.

Advantages of WebGL

3

Browser (Chrome)

Figure 1.2

Notepad

The only tools needed for developing 3D graphics applications using WebGL

Publishing Your 3D Graphics Applications Is Easy
Traditionally, 3D graphics applications have been developed using a programming
language such as C or C++ and then compiled into an executable binary for a specific platform. This meant, for example, the version for a Macintosh wouldn’t work on Windows or
Linux. Additionally, users often needed to install not only the applications themselves but
also libraries required by the applications to run, which meant another level of complexity
when you wanted to share your work.
In contrast, because WebGL applications are composed of HTML and JavaScript files,
they can be easily shared by simply putting them on a web server just like standard web
pages or distributing the HTML and JavaScript files via email. For example, Figure 1.3
shows some sample WebGL applications published by Google and available at http://code.
google.com/p/webglsamples/.

Figure 1.3 WebGL sample applications published by Google (with the permission of Gregg
Tavares, Google)
4

CHAPTER 1

Overview of WebGL

You Can Leverage the Full Functionality of the Browser
Because WebGL applications are created as part of a web page, you can utilize the full
functionality of the browser such as arranging buttons, displaying dialog boxes, drawing
text, playing video or audio, and communicating with web servers. These advanced
features come for free, whereas in traditional 3D graphics applications they would need to
be programmed explicitly.

Learning and Using WebGL Is Easy
The specification of WebGL is based on the royalty-free open standard, OpenGL, which
has been widely used in graphics, video games, and CAD applications for many years. In
one sense, WebGL is “OpenGL for web browsers.” Because OpenGL has been used in a
variety of platforms over the past 20 years, there are many reference books, materials, and
sample programs using OpenGL, which can be used to better understand WebGL.

Origins of WebGL
Two of the most widely used technologies for displaying 3D graphics on personal computers are Direct3D and OpenGL. Direct3D, which is part of Microsoft’s DirectX technologies,
is the 3D graphics technology primarily used on Windows platforms and is a proprietary
application programming interface (API) that Microsoft controls. An alternative, OpenGL
has been widely used on various platforms due to its open and royalty-free nature.
OpenGL is available for Macintosh, Linux, and a variety of devices such as smart phones,
tablet computers, and game consoles (PlayStation and Nintendo). It is also well supported
on Windows and provides an alternative to Direct3D.
OpenGL was originally developed by Silicon Graphics Inc. and published as an open
standard in 1992. OpenGL has evolved through several versions since 1992 and has had a
profound effect on the development of 3D graphics, software product development, and
even film production over the years. The latest version of OpenGL at the time of writing
is version 4.3 for desktop PCs. Although WebGL has its roots in OpenGL, it is actually
derived from a version of OpenGL designed specifically for embedded computers such as
smart phones and video game consoles. This version, known as OpenGL ES (for Embedded
Systems), was originally developed in 2003–2004 and was updated in 2007 (ES 2.0) and
again in 2012 (ES 3.0). WebGL is based on the ES 2.0 version. In recent years, the number
of devices and processors that support the specification has rapidly increased and includes
smart phones (iPhone and Android), tablet computers, and game consoles. Part of the
reason for this successful adoption has been that OpenGL ES added new features but also
removed many unnecessary or old-fashioned features from OpenGL, resulting in a lightweight specification that still had enough visual expressive power to realize attractive 3D
graphics.
Figure 1.4 shows the relationship among OpenGL, OpenGL ES 1.1/2.0/3.0, and WebGL.
Because OpenGL itself has continued to evolve from 1.5, to 2.0, to 4.3, OpenGL ES
have been standardized as a subset of specific versions of OpenGL (OpenGL 1.5 and
OpenGL 2.0).
Origins of WebGL

5

OpenGL 1.5

OpenGL 2.0

OpenGL 3.3

OpenGL ES
1.1

OpenGL ES
2.0

OpenGL ES
3.0

OpenGL 4.3

WebGL 1.0

Figure 1.4

Relationship among OpenGL, OpenGL ES 1.1/2.0/3.0, and WebGL

As shown in Figure 1.4, with the move to OpenGL 2.0, a significant new capability,
programmable shader functions, was introduced. This capability has been carried
through to OpenGL ES 2.0 and is a core part of the WebGL 1.0 specification.
Shader functions or shaders are computer programs that make it possible to program
sophisticated visual effects by using a special programming language similar to C. This
book explains shader functions in a step-by-step manner, allowing you to quickly master
the power of WebGL. The programming language that is used to create shaders is called
a shading language. The shading language used in OpenGL ES 2.0 is based on the
OpenGL shading language (GLSL) and referred to as OpenGL ES shading language
(GLSL ES). Because WebGL is based on OpenGL ES 2.0, it also uses GLSL ES for creating
shaders.
The Khronos Group (a non-profit industry consortium created to develop, publish, and
promote various open standards) is responsible for the evolution and standardization of
OpenGL. In 2009, Khronos established the WebGL working group and then started the
standardization process of WebGL based on OpenGL ES 2.0, releasing the first version of
WebGL in 2011. This book is written based primarily on that specification and, where
needed, the latest specification of WebGL published as an Editor’s Draft. For more information, please refer to the specification.1

Structure of WebGL Applications
In HTML, dynamic web pages can be created by using a combination of HTML and
JavaScript. With the introduction of WebGL, the shader language GLSL ES needs to
be added to the mix, meaning that web pages using WebGL are created by using three

1

6

WebGL 1.0: www.khronos.org/registry/webgl/specs/1.0/ and Editor’s draft: www.khronos.org/
registry/webgl/specs/latest/

CHAPTER 1

Overview of WebGL

languages: HTML5 (as a Hypertext Markup Language), JavaScript, and GLSL ES. Figure 1.5
shows the software architecture of traditional dynamic web pages (left side) and web pages
using WebGL (right side).

Web page

HTML

JavaScript

HTML
Rendering Engine

Web page using WebGL

HTML5

JavaScript

HTML
HTML

GLSL ES
WebGL

Rendering Engine
OpenGL/OpenGL ES

Figure 1.5 The software architecture of dynamic web pages (left) and web pages using
WebGL (right)
However, because GLSL ES is generally written within JavaScript, only HTML and
JavaScript files are actually necessary for WebGL applications. So, although WebGL does
add complexity to the JavaScript, it retains the same structure as standard dynamic web
pages, only using HTML and JavaScript files.

Summary
This chapter briefly overviewed WebGL, explained some key features, and outlined
the software architecture of WebGL applications. In summary, the key takeaway from
this chapter is that WebGL applications are developed using three languages: HTML5,
JavaScript, and GLSL ES—however, because the shader code (GLSL ES) is generally embedded in the JavaScript, you have exactly the same file structure as a traditional web page.
The next chapter explains how to create applications using WebGL, taking you step by
step through a set of simple WebGL examples.

Summary

7

This page intentionally left blank

Chapter 2

Your First Step with WebGL

As explained in Chapter 1, “Overview of WebGL,” WebGL applications use both HTML and
JavaScript to create and draw 3D graphics on the screen. To do this, WebGL utilizes the new
 element, introduced in HTML5, which defines a drawing area on a web page. Without
WebGL, the  element only allows you to draw two-dimensional graphics using
JavaScript. With WebGL, you can use the same element for drawing three-dimensional graphics.
This chapter explains the  element and the core functions of WebGL by taking you,
step-by-step, through the construction of several example programs. Each example is written in
JavaScript and uses WebGL to display and interact with a simple shape on a web page. Because of
this, these JavaScript programs are referred to as WebGL applications.
The example WebGL applications will highlight some key points, including:

• How WebGL uses the  element and how to draw on it
• The linkage between HTML and WebGL using JavaScript
• Simple WebGL drawing functions
• The role of shader programs within WebGL
By the end of this chapter, you will understand how to write and execute basic WebGL applications and how to draw simple 2D shapes. You will use this knowledge to explore further the
basics of WebGL in Chapters 3, “Drawing and Transforming Triangles,” 4, “More Transformations
and Basic Animation,” and 5, “Using Colors and Texture Images.”

What Is a Canvas?
Before HTML5, if you wanted to display an image in a web page, the only native HTML approach
was to use the  tag. This tag, although a convenient tool, is restricted to still images and

doesn’t allow you to dynamically draw and display the image on the fly. This is one of the
reasons that non-native solutions such as Flash Player have been used.
However, HTML5, by introducing the  tag, has changed all that, offering a convenient way to draw computer graphics dynamically using JavaScript.
In a similar manner to the way artists use paint canvases, the  tag defines a
drawing area on a web page. Then, rather than using brush and paints, you can use
JavaScript to draw anything you want in the area. You can draw points, lines, rectangles,
circles, and so on by using JavaScript methods provided for . Figure 2.1 shows an
example of a drawing tool that uses the  tag.

Figure 2.1

A drawing tool using the  element (http://caimansys.com/painter/)

This drawing tool runs within a web page and allows you to interactively draw lines, rectangles, and circles and even change their colors.
Although you won’t be creating anything as sophisticated just yet, let’s look at the core
functions of  by using a sample program, DrawRectangle, which draws a filled
blue rectangle on a web page. Figure 2.2 shows DrawRectangle when it’s loaded into a
browser.

10

CHAPTER 2

Your First Step with WebGL

Figure 2.2

DrawRectangle

Using the  Tag
Let’s look at how DrawRectangle works and explain how the  tag is used in the
HTML file. Listing 2.1 shows DrawingTriangle.html. Note that all HTML files in this book
are written in HTML5.
Listing 2.1
1
2
3
4
5
6
7
8
9
10
11
12
13
14

DrawRectangle.html









Please use a browser that supports "canvas"





What Is a Canvas?

11

The  tag is defined at line 9. This defines the drawing area as a 400 × 400 pixel
region on a web page using the width and height attributes in the tag. The canvas is given
an identifier using the id attribute, which will be used later:


By default, the canvas is invisible (actually transparent) until you draw something into it,
which we’ll do with JavaScript in a moment. That’s all you need to do in the HTML file to
prepare a  that the WebGL program can use. However, one thing to note is that
this line only works in a -enabled browser. However, browsers that don’t support
the  tag will ignore this line, and nothing will be displayed on the screen. To
handle this, you can display an error message by adding the message into the tag as
follows:
9
10
11


Please use a browser that supports "canvas"


To draw into the canvas, you need some associated JavaScript code that performs the
drawing operations. You can include that JavaScript code in the HTML or write it as a
separate JavaScript file. In our examples, we use the second approach because it makes the
code easier to read. Whichever approach you take, you need to tell the browser where the
JavaScript code starts. Line 8 does that by telling the browser that when it loads the separate JavaScript code it should use the function main() as the entry point for the JavaScript
program. This is specified for the  element using its onload attribute that tells the
browser to execute the JavaScript function main() after it loads the  element:
8



Line 12 tells the browser to import the JavaScript file DrawRectangle.js in which the
function main() is defined:
12



For clarity, all sample programs in this book use the same filename for both the HTML file
and the associated JavaScript file, which is imported in the HTML file (see Figure 2.3).

12

CHAPTER 2

Your First Step with WebGL

< !D OC T YPE h t ml>



< t i tl e >D r a w a b l u e r e c ta n gl e ( c a nv a s v e r si o n)< / t i t l e >
< / h ea d >


P le a se us e a br o w s e r t h a t su p p o r t s “ c a n v a s ”
< /ca nva s>


< /h t m l >

DrawRectangle.html

/ / D ra wR ectangl e.js
f un c t i on m ai n ( ) {
/ / R e t r ie ve t h e < c a n v a s > e l em e n t
var canvas = document.getElementById('example');
i f (! canv as) {
c o n sole .l og( Fa il e d to retr ieve t he < c an vas > e lemen t ' );
re tu rn f al se ;
}
/ / G e t t h e r e n de r i n g co nt ex t f or 2 D C G
v a r c tx = canva s.getC ontex t(' 2d ');
// Draw a blue rectangle
c tx . f i l l S t y l e = ' r g b a ( 0 , 0 , 2 5 5 , 1 . 0 ) ' ; / / S e t a b l u e c o l o r
ct x.fi ll Rec t(12 0, 10, 150, 15 0);
// Fill a rec tang le wi th the c olor
}

DrawRectangle.js
Figure 2.3

DrawRectangle.html and DrawRectangle.js

DrawRectangle.js
DrawRectangle.js is a JavaScript program that draws a blue rectangle on the drawing area
defined by the  element (see Listing 2.2). It has only 16 lines, which consist of

the three steps required to draw two-dimensional computer graphics (2D graphics) on the
canvas:
1. Retrieve the  element.
2. Request the rendering “context” for the 2D graphics from the element.
3. Draw the 2D graphics using the methods that the context supports.
These three steps are the same whether you are drawing a 2D or a 3D graphic; here, you
are drawing a simple 2D rectangle using standard JavaScript. If you were drawing a 3D
graphic using WebGL, then the rendering context in step (2) at line 11 would be for a 3D
rendering context; however, the high-level process would be the same.
Listing 2.2
1
2
3
4
5
6
7
8
9
10
11

DrawRectangle.js

// DrawRectangle.js
function main() {
// Retrieve  element
var canvas = document.getElementById('example');
if (!canvas) {
console.log('Failed to retrieve the  element');
return;
}
// Get the rendering context for 2DCG
var ctx = canvas.getContext('2d');

<- (1)

<- (2)

What Is a Canvas?

13

12
13
14
15
16

// Draw a blue rectangle
<- (3)
ctx.fillStyle = 'rgba(0, 0, 255, 1.0)'; // Set a blue color
ctx.fillRect(120, 10, 150, 150); // Fill a rectangle with the color
}

Let us look at each step in order.
Retrieve the  Element
To draw something on a , you must first retrieve the  element from the
HTML file in the JavaScript program. You can get the element by using the method document.getElementById(), as shown at line 4. This method has a single parameter, which is
the string specified in the attribute id in the  tag in our HTML file. In this case,
the string is 'example' and it was defined back in DrawRectangle.html at line 9 (refer to
Listing 2.1).
If the return value of this method is not null, you have successfully retrieved the element.
However, if it is null, you have failed to retrieve the element. You can check for this
condition using a simple if statement like that shown at line 5. In case of error, line 6 is
executed. It uses the method console.log() to display the parameter as a string in the
browser’s console.
Note In Chrome, you can show the console by going to Tools, JavaScript Console or
pressing Ctrl+Shift+J (see Figure 2.4); in Firefox, you can show it by going to Tools, Web
Developer, Web Console or pressing Ctrl+Shift+K.

Console

Figure 2.4

14

Console in Chrome

CHAPTER 2

Your First Step with WebGL

Get the Rendering Context for the 2D Graphics by Using the Element
Because the  is designed to be flexible and supports both 2D and 3D, it does not
provide drawing methods directly and instead provides a mechanism called a context,
which supports the actual drawing features. Line 11 gets that context:
11

var ctx = canvas.getContext('2d');

The method canvas.getContext() has a parameter that specifies which type of drawing
features you want to use. In this example you want to draw a 2D shape, so you must
specify 2d (case sensitive).
The result of this call, the context, is stored in the variable ctx ready for use. Note, for
brevity we haven’t checked error conditions, which is something you should always do in
your own programs.
Draw the 2D Graphics Using the Methods Supported by the Context
Now that we have a drawing context, let’s look at the code for drawing a blue rectangle,
which is a two-step process. First, set the color to be used when drawing. Second, draw (or
fill) a rectangle with the color.
Lines 14 and 15 handle these steps:
13
14
15

// Draw a blue rectangle
<- (3)
ctx.fillStyle = 'rgba(0, 0, 255, 1.0)'; // Set color to blue
ctx.fillRect(120, 10, 150, 150); // Fill a rectangle with the color

The rgba in the string rgba(0, 0, 255, 1.0) on line 14 indicate r (red), g (green), b
(blue), and a (alpha: transparency), with each RGB parameter taking a value from 0
(minimum value) to 255 (maximum value) and the alpha parameter from 0.0 (transparent) to 1.0 (opaque). In general, computer systems represent a color by using a combination of red, green, and blue (light’s three primary colors), which is referred to as RGB
format. When alpha (transparency) is added, the format is called RGBA format.
Line 15 then uses the fillStyle property to specify the fill color when drawing the rectangle. However, before going into the details of the arguments on line 15, let’s look at the
coordinate system used by the  element (see Figure 2.5).

What Is a Canvas?

15

x

(0, 0)
(120, 10)

y


drawing area

(400, 400)

Figure 2.5

The coordinate system of 

As you can see in the figure, the coordinate system of the  element has the horizontal direction as the x-axis (right-direction is positive) and the vertical direction as the
y-axis (down-direction is positive). Note that the origin is located at the upper-left corner
and the down direction of the y-axis is positive. The rectangle drawn with a dashed line
is the original  element in our HTML file (refer to Listing 2.1), which we specified as being 400 by 400 pixels. The dotted line is the rectangle that the sample program
draws.
When we use ctx.fillRect() to draw a rectangle, the first and second parameters of this
method are the position of the upper-left corner of the rectangle within the , and
the third and fourth parameters are the width and height of the rectangle (in pixels):
15

ctx.fillRects(120, 10, 150, 150);// Fill a rectangle with the color

After loading DrawRectangle.html into your browser, you will see the rectangle that was
shown in Figure 2.2.
So far, we’ve only looked at 2D graphics. However, WebGL also utilizes the same 
element to draw 3D graphics on a web page, so let’s now enter into the WebGL world.

The World’s Shortest WebGL Program:
Clear Drawing Area
Let’s start by constructing the world’s shortest WebGL program, HelloCanvas, which
simply clears the drawing area specified by a  tag. Figure 2.6 shows the result of
loading the program, which clears (by filling with black) the rectangular area defined by a
.

16

CHAPTER 2

Your First Step with WebGL

Figure 2.6

HelloCanvas

The HTML File (HelloCanvas.html)
Take a look at HelloCanvas.html, as shown in Figure 2.7). Its structure is simple and starts
by defining the drawing area using the  element at line 9 and then importing
HelloCanvas.js (the WebGL program) at line 16.
Lines 13 to 15 import several other JavaScript files, which provide useful convenience
functions that help WebGL programming. These will be explained in more detail later. For
now, just think of them as libraries.

The World’s Shortest WebGL Program: Clear Drawing Area

17

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18

< !D O C T Y P E h tm l>
< h tm l l a n g = ” e n ” >

< m e ta c h a rs e t= "u tf-8 " />
< t it le > C le a r c a n v a s < / t it le >
< /h e a d >
< b o d y o n lo a d = " m a in ( ) " >
< c a n v a s id = " w e b g l" w id t h = " 4 0 0 " h e ig h t= " 4 0 0 " >
P le a s e u s e th e b r o w s e r s u p p o r tin g " c a n v a s "
< /c a n v a s >
< s c r ip t
< s c r ip t
< s c r ip t
< s c r ip t
< /b o d y >
< /h tm l>

Figure 2.7

s r c = " ../lib /w e b g l-u tils .js " > < /s c r ip t>
s r c = " ../lib /w e b g l-d e b u g .js " > < /s c r ip t>
s r c = " ../lib /c u o n -u tils .js " > < /s c r ip t>
s r c = " H e llo C a n v a s .js " > < /s c r ip t>

< c a n v a s > in to w h ic h
W e b G L d ra w s s h a p e s

J a v a S c r ip t file s
c o n ta in in g c o n v e n ie n t
fu n c tio n s fo r W e b G L
J a v a S c r ip t file d r a w in g
s h a p e s in to th e < c a n v a s >

HelloCanvas.html

You’ve set up the canvas (line 9) and then imported the HelloCanvas JavaScript file (line
16), which actually uses WebGL commands to access the canvas and draw your first 3D
program. Let us look at the WebGL program defined in HelloCanvas.js.

JavaScript Program (HelloCanvas.js)
HelloCanvas.js (see Listing 2.3) has only 18 lines, including comments and error
handling, and follows the same steps as explained for 2D graphics: retrieve the 

element, get its rendering context, and then begin drawing.
Listing 2.3

HelloCanvas.js

1 // HelloCanvas.js
2 function main() {
3
// Retrieve  element
4
var canvas = document.getElementById('webgl');
5
6
// Get the rendering context for WebGL
7
var gl = getWebGLContext(canvas);
8
if (!gl) {
9
console.log('Failed to get the rendering context for WebGL');
10
return;
11
}

18

CHAPTER 2

Your First Step with WebGL

12
13
14
15
16
17
18 }

// Specify the color for clearing 
gl.clearColor(0.0, 0.0, 0.0, 1.0);
// Clear 
gl.clear(gl.COLOR_BUFFER_BIT);

As in the previous example, there is only one function, main(), which is the link between
the HTML and the JavaScript and set at  element using its onload attribute (line 8)
in HelloCanvas.html (refer to Figure 2.7).
Figure 2.8 shows the processing flow of the main() function of our WebGL program and
consists of four steps, which are discussed individually next.

Retrieve the  element

Get the rendering context for WebGL

Set the color clearing 

Clear 
Figure 2.8

The processing flow of the main() function

Retrieve the  Element
First, main() retrieves the  element from the HTML file. As explained in
DrawRectangle.js, it uses the document.getElementById() method specifying webgl as the
argument. Looking back at HelloCanvas.html (refer to Figure 2.7), you can see that attribute id is set at the  tag at line 9:
9



The return value of this method is stored in the canvas variable.

The World’s Shortest WebGL Program: Clear Drawing Area

19

Get the Rendering Context for WebGL
In the next step, the program uses the variable canvas to get the rendering context
for WebGL. Normally, we would use canvas.getContext() as described earlier to
get the rendering context for WebGL. However, because the argument specified in
canvas.getContext() varies between browsers,1 we have written a special function
getWebGLContext() to hide the differences between the browsers:
7

var gl = getWebGLContext(canvas);

This is one of the convenience functions mentioned earlier that was written specially for
this book and is defined in cuon-utils.js, which is imported at line 15 in HelloCanvas.
html. The functions defined in the file become available by specifying the path to the file
in the attribute src in the 
src="../libs/webgl-debug.js">
src="../libs/cuon-utils.js">
src="HelloPoint1.js">

HelloPoint1.js
Listing 2.5 shows HelloPoint1.js. As you can see from the comments in the listing,
two “shader programs” are prepended to the JavaScript (lines 2 to 13). Glance through
the shader programs, and then go to the next section, where you’ll see more detailed
explanations.

Draw a Point (Version 1)

25

Listing 2.5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40

26

HelloPoint1.js

// HelloPoint1.js
// Vertex shader program
var VSHADER_SOURCE =
'void main() {\n' +
' gl_Position = vec4(0.0, 0.0, 0.0, 1.0);\n' + // Coordinates
' gl_PointSize = 10.0;\n' +
// Set the point size
'}\n';
// Fragment shader program
var FSHADER_SOURCE =
'void main() {\n' +
' gl_FragColor = vec4(1.0, 0.0, 0.0, 1.0);\n' + // Set the color
'}\n';
function main() {
// Retrieve  element
var canvas = document.getElementById('webgl');
// Get the rendering context for WebGL
var gl = getWebGLContext(canvas);
if (!gl) {
console.log('Failed to get the rendering context for WebGL');
return;
}
// Initialize shaders
if (!initShaders(gl, VSHADER_SOURCE, FSHADER_SOURCE)) {
console.log('Failed to initialize shaders.');
return;
}
// Set the color for clearing 
gl.clearColor(0.0, 0.0, 0.0, 1.0);
// Clear 
gl.clear(gl.COLOR_BUFFER_BIT);
// Draw a point
gl.drawArrays(gl.POINTS, 0, 1);
}

CHAPTER 2

Your First Step with WebGL

What Is a Shader?
HelloPoint1.js is our first WebGL program that uses shaders. As mentioned earlier,
shader programs are necessary when you want to draw something on the screen in
WebGL. Essentially, shader programs are “embedded” in the JavaScript file and, in this
case, set up at the start. This seems at first sight to be complicated, but let’s take it one
step at a time.

WebGL needs the following two types of shaders, which you saw at line 2 and line 9:

• Vertex shader: Vertex shaders are programs that describe the traits (position, colors,
and so on) of a vertex. The vertex is a point in 2D/3D space, such as the corner or
intersection of a 2D/3D shape.

• Fragment shader: A program that deals with per-fragment processing such as lighting (see Chapter 8, “Lighting Objects”). The fragment is a WebGL term that you can
consider as a kind of pixel (picture element).
You’ll explore shaders in more detail later, but simply put, in a 3D scene, it’s not enough
just to draw graphics. You have to also account for how they are viewed as light sources
hit them or the viewer’s perspective changes. Shading does this with a high degree of flexibility and is part of the reason that today’s 3D graphics are so realistic, allowing them to
use new rendering effects to achieve stunning results.
The shaders are read from the JavaScript and stored in the WebGL system ready to be used
for drawing. Figure 2.10 shows the basic processing flow from a JavaScript program into
the WebGL system, which applies the shader programs to draw shapes that the browser
finally displays.

JavaScript program is executed (onload)

JavaScript

WebGL-related methods are executed
Per-vertex operation

Vertex Shader

Browser

y

Per-fragment operation

Fragment
Shader

function main() {
var gl=getWebGL…
…
i nitShaders ( … ) ;
…
}

x
P osi tion:
(0.0, 0.0, 0.0 , 1.0)
S ize : 1 0. 0

Color:
(1.0, 0.0, 0.0, 1.0)

W e bG L S yste m
Render to the color buffer

Display
Color Buffer

Figure 2.10
in a browser

The processing flow from executing a JavaScript program to displaying the result

Draw a Point (Version 1)

27

You can see two browser windows on the left side of the figure. These are the same; the
upper one shows the browser before executing the JavaScript program, and the lower one
shows the browser after execution. Once the WebGL-related methods are called from the
JavaScript program, the vertex shader in the WebGL system is executed, and the fragment
shader is executed to draw the result into the color buffer. This is the clear part—that is,
step 4 in Figure 2.8, described in the original HelloCanvas example. Then the content in
the color buffer is automatically displayed on the drawing area specified by the 
in the browser.
You’ll be seeing this figure frequently in the rest of this book. So we’ll use a simplified
version to save space (see Figure 2.11). Note that the flow is left to right and the rightmost component is a color buffer, not a browser, because the color buffer is automatically
displayed in the browser.
Per-vertex operation

Per-fragment operation

JavaScript
Fragment
Shader

Vertex Shader
function main() {

y

var gl=getWebGL…
…
initShaders(…);
…
}

x
Po si ti on:
(0.0, 0.0, 0.0, 1.0)
S ize: 10.0

Color:
(1.0, 0.0, 0.0, 1.0)

Color Buffer

WebGL System
Figure 2.11

The simplified version of Figure 2.9

Getting back to our example, the goal is to draw a 10-pixel point on the screen. The two
shaders are used as follows:

• The vertex shader specifies the position of a point and its size. In this sample
program, the position is (0.0, 0.0, 0.0), and the size is 10.0.

• The fragment shader specifies the color of fragments displaying the point. In this
sample program, the color is red (1.0, 0.0, 0.0, 1.0).

The Structure of a WebGL Program that Uses Shaders
Based on what you’ve learned so far, let’s look at HelloPoint1.js again (refer to Listing
2.5). This program has 40 lines and is a little more complex than HelloCanvas.js (18
lines). It consists of three parts, as shown in Figure 2.12. The main() function in JavaScript
starts from line 15, and shader programs are located from lines 2 to 13.

28

CHAPTER 2

Your First Step with WebGL

1 / / H e llo P in t 1 . js
2 // V e rte x s h a d e r p ro g ra m
3 var V S H A D E R _S O U R C E =
4 ‘ v o id m a in ( ) { \ n ' +
5 ' g l_ P o s it io n = v e c 4 ( 0 . 0 , 0 . 0 , 0 . 0 , 1 . 0 ) ; \ n ' +

Vertex shader program
(Language: GLSL ES)

6 ' g l_ P o in t S iz e = 1 0 . 0 ; \ n ' +
7 '} \ n ';
8
9 // F ra g m e n t s h a d e r p ro g ra m
10 v ar FSH AD ER _SO U R C E =
1 1 'v o id m a in ( ) { \ n ' +

Fragment shader program
(Language: GLSL ES)

1 2 ' g l_ F r a g C o lo r = v e c 4 ( 1 . 0 , 0 . 0 , 0 . 0 , 1 . 0 ) ; \ n ' +
1 3 '} \ n ';
14
1 5 f u n c t io n m a in ( ) {
1 6 // R e t r ie v e < c a n v a s > e le m e n t
1 7 v a r c a n v a s = d o c u m e n t . g e t E le m e n t B y I d ( 'w e b g l ' ) ;
18
1 9 / / G e t t h e r e n d e r in g c o n t e x t f o r W e b G L
2 0 v a r g l = g e tW e b G L C o n te x t(c a n v a s ) ;
…
2 6 / / I n it ia liz e s h a d e r s
2 7 if ( ! i n it S h a d e r s ( g l, V S H A D E R _ S O U R C E , …
…
3 2 / / S e t t h e c o lo r f o r c le a r in g < c a n v a s >

Main program
(Language: JavaScript)

3 3 g l. c le a r C o lo r ( 0 . 0 , 0 . 0 , 0 . 0 , 1 . 0 ) ;
…
3 5 / / C le a r < c a n v a s >
3 6 g l . c le a r ( g l. C O L O R _ B U F F E R _ B I T ) ;
37
3 8 // D r a w a p o in t
3 9 g l. d r a w A r r a y s ( g l. P O I N T S , 0 , 1 ) ;
40 }

Figure 2.12

The basic structure of a WebGL program with embedded shader programs

The vertex shader program is located in lines 4 to 7, and the fragment shader is located in
lines 11 to 13. These programs are actually the following shader language programs but
written as a JavaScript string to make it possible to pass the shaders to the WebGL system:
// Vertex shader program
void main() {
gl_Position = vec4(0.0, 0.0, 0.0, 1.0);
gl_PointSize = 10.0;
}
// Fragment shader program
void main() {
gl_FragColor = vec4(1.0, 0.0, 0.0, 1.0);
}

Draw a Point (Version 1)

29

As you learned in Chapter 1, shader programs must be written in the OpenGL ES shading
language (GLSL ES), which is similar to the C language. Finally, GLSL ES comes onto the
stage! You will get to see the details of GLSL ES in Chapter 6, “The OpenGL ES Shading
Language (GLSL ES),” but in these early examples, the code is simple and should be understandable by anybody with a basic understanding of C or JavaScript.
Because these programs must be treated as a single string, each line of the shader is
concatenated using the + operator into a single string. Each line has \n at the end because
the line number is displayed when an error occurs in the shader. The line number is
helpful to check the source of the problem in the codes. However, the \n is not mandatory, and you could write the shader without it.
At lines 3 and 10, each shader program is stored in the variables VSHADER_SOURCE and
FSHADER_SOURCE as a string:
2
3
4
5
6
7
8
9
10
11
12
13

// Vertex shader program
var VSHADER_SOURCE =
'void main() {\n' +
' gl_Position = vec4(0.0, 0.0, 0.0, 1.0);\n' +
' gl_PointSize = 10.0;\n' +
'}\n';
// Fragment shader program
var FSHADER_SOURCE =
'void main() {\n' +
' gl_FragColor = vec4(1.0, 0.0, 0.0, 1.0);\n' +
'}\n';

If you are interested in loading the shader programs from files, refer to Appendix F,
“Loading Shader Programs from Files.”

Initializing Shaders
Before looking at the details of each shader, let’s examine the processing flow of main()
that is defined from line 15 in the JavaScript (see Figure 2.13). This flow, shown in Figure
2.13, is the basic processing flow of most WebGL applications. You will see the same flow
throughout this book.

30

CHAPTER 2

Your First Step with WebGL

Retrieve the  element

Get the rendering context for WebGL

Initialize shaders

Set the color for clearing 

Clear 

Draw

Figure 2.13

The processing flow of a WebGL program

This flow is similar to that shown in Figure 2.8 except that a third step (“Initialize
Shaders”) and a sixth step (“Draw”) are added.
The third step “Initialize Shaders” initializes and sets up the shaders that are defined at
line 3 and line 10 within the WebGL system. This step is done using the convenience
function initShaders() that is defined in cuon-util.js. Again, this is one of those special
functions we have provided for this book.

initShaders(gl, vshader, fshader)
Initialize shaders and set them up in the WebGL system ready for use:
Parameters

Return value

gl

Specifies a rendering context.

vshader

Specifies a vertex shader program (string).

fshader

Specifies a fragment shader program (string).

true

Shaders successfully initialized.

false

Failed to initialize shaders.

Figure 2.14 shows how the convenience function initShaders() processes the shaders.
You will examine in detail what this function is doing in Chapter 8. For now, you just
need to understand that it sets up the shaders in the WebGL system and makes them
ready for use.
Draw a Point (Version 1)

31

Per-vertex operation

Color Buffer

Per-fragment operation

JavaScript

Vertex Shader

Fragment
Shader

void main() {
gl_FragColor = vec4(...);
}
fu n c tio n m a in ( ) {
v a r g l= g e t W e b G L …
…
i n itS h a d e r s ( … ) ;
…
}

Figure 2.14

void main() {
gl_Position = vec4(...);
gl_PointSize = 10.0;
}

Fragment
Shader
void main() {
gl_FragColor = vec4(...);
}

gl_FragColor

Vertex Shader

gl_PointSize

JavaScript
void main() {
gl_Position = vec4(...);
gl_PointSize = 10.0;
}

gl_Position

W e bG L S yste m

W e bG L S yste m

Behavior of initShaders()

As you can see in the upper figure in Figure 2.14, the WebGL system has two containers:
one for a vertex shader and one for a fragment shader. This is actually a simplification,
but helpful at this stage. We return to the details in Chapter 10. By default, the contents
of these containers are empty. To make the shader programs, written as JavaScript strings
and ready for use in the WebGL system, we need something to pass these strings to the
system and then set them up in the appropriate containers; initShaders() performs this
operation. Note that the shader programs are executed within the WebGL system, not the
JavaScript program.
The lower portion in Figure 2.14 shows that after executing initShaders(), the shader
programs that are passed as a string to the parameters of initShaders() are set up in the
containers in the WebGL system and then made ready for use. The lower figure schematically illustrates that a vertex shader is passing gl_Position and gl_PointSize to a fragment shader and that just after assigning values to these variables in the vertex shader, the
fragment shader is executed. In actuality, the fragments that are generated after processing these values are passed to the fragment shader. Chapter 5 explains this mechanism in
detail, but for now you can consider the attributes to be passed.

32

CHAPTER 2

Your First Step with WebGL

The important point here is that WebGL applications consist of a JavaScript program executed
by the browser and shader programs that are executed within the WebGL system.
Now, having completed the explanation of the second step “Initialize Shaders” in Figure
2.13, you are ready to see how the shaders are actually used to draw a simple point. As
mentioned, we need three items of information for the point: its position, size, and color,
which are used as follows:

• The vertex shader specifies the position of a point and its size. In this sample
program, the position is (0.0, 0.0, 0.0), and the size is 10.0.

• The fragment shader specifies the color of the fragments displaying the point. In this
sample program, they are red (1.0, 0.0, 0.0, 1.0).

Vertex Shader
Now, let us start by examining the vertex shader program listed in HelloPoint1.js (refer
to Listing 2.5), which sets the position and size of the point:
2
3
4
5
6
7

// Vertex shader program
var VSHADER_SOURCE =
'void main() {\n' +
' gl_Position = vec4(0.0, 0.0, 0.0, 1.0);\n' +
' gl_PointSize = 10.0;\n' +
'}\n';

The vertex shader program itself starts from line 4 and must contain a single main() function in a similar fashion to languages such as C. The keyword void in front of main()
indicates that this function does not return a value. You cannot specify other arguments
to main().
Just like JavaScript, we can use the = operator to assign a value to a variable in a shader.
Line 5 assigns the position of the point to the variable gl_Position, and line 6 assigns its
size to the variable gl_PointSize. These two variables are built-in variables available only
in a vertex shader and have a special meaning: gl_Position specifies a position of a vertex
(in this case, the position of the point), and gl_PointSize specifies the size of the point
(see Table 2.2).
Table 2.2

Built-In Variables Available in a Vertex Shader

Type and Variable Name

Description

vec4 gl_Position

Specifies the position of a vertex

float gl_PointSize

Specifies the size of a point (in pixels)

Draw a Point (Version 1)

33

Note that gl_Position should always be written. If you don’t specify it, the shader’s
behavior is implementation dependent and may not work as expected. In contrast, gl_
PointSize is only required when drawing points and defaults to a point size of 1.0 if you
don’t specify anything.
For those of you mostly familiar with JavaScript, you may be a little surprised when you
see “type” specified in Table 2.2. Unlike JavaScript, GLSL ES is a “typed” programming
language; that is, it requires the programmer to specify what type of data a variable holds.
C and Java are examples of typed languages. By specifying “type” for a variable, the
system can easily understand what type of data the variable holds, and then it can optimize its processing based on that information. Table 2.3 summarizes the “type” in GLSL
ES used in the shaders in this section.
Table 2.3

Data Types in GLSL ES

Type

Description

float

Indicates a floating point number

vec4

Indicates a vector of four floating point numbers
float

float

Float

float

Note that an error will occur when the type of data that is assigned to the variable is
different from the type of the variable. For example, the type of gl_PointSize is float, and
you must assign a floating point number to it. So, if you change line 6 from
gl_PointSize = 10.0;

to
gl_PointSize = 10;

it will generate an error simply because 10 is interpreted as an integer number, whereas
10.0 is a floating point number in GLSL ES.
The type of the variable gl_Position, the built-in variable for specifying the position of
a point, is vec4; vec4 is a vector made up of three floats. However, you only have three
floats (0.0, 0.0, 0.0) representing X, Y, and Z. So you need to convert these to a vec4
somehow. Fortunately, there is a built-in function, vec4(), that will do this for you and
return a value of type vec4 –, which is just what you need!

34

CHAPTER 2

Your First Step with WebGL

vec4 vec4(v0, v1, v2, v3)
Construct a vec4 object from v0, v1, v2, and v3.
Parameters

v0, v1, v2, v3

Specifies floating point numbers.

Return value

A vec4 object made from v0, v1, v2, and v3.

In this sample program, vec4() is used at line 5 as follows:
gl_Position = vec4(0.0, 0.0, 0.0, 1.0);

Note that the value that is assigned to gl_Position has 1.0 added as a fourth component.
This four-component coordinate is called a homogeneous coordinate (see the boxed
article below) and is often used in 3D graphics for processing three-dimensional information efficiently. Although the homogeneous coordinate is a four-dimensional coordinate,
if the last component of the homogeneous coordinate is 1.0, the coordinate indicates the
same position as a three-dimensional one. So, you can supply 1.0 as the last component if
you need to specify four components as a vertex coordinate.

Homogeneous Coordinates
The homogeneous coordinates use the following coordinate notation: (x, y, z, w). The
homogeneous coordinate (x, y, z, w) is equivalent to the three-dimensional coordinate
(x/w, y/w, z/w). So, if you set w to 1.0, you can utilize the homogeneous coordinate
as a three-dimensional coordinate. The value of w must be greater than or equal to
0. If w approaches zero, the coordinates approach infinity. So we can represent the
concept of infinity in the homogeneous coordinate system. Homogeneous coordinates
make it possible to represent vertex transformations described in the next chapter as a
multiplication of a matrix and the coordinates. These coordinates are often used as an
internal representation of a vertex in 3D graphics systems.

Fragment Shader
After specifying the position and size of a point, you need to specify its color using a fragment shader. As explained earlier, a fragment is a pixel displayed on the screen, although
technically the fragment is a pixel along with its position, color, and other information.
The fragment shader is a program that processes this information in preparation for
displaying the fragment on the screen. Looking again at the fragment shader listed in
HelloPoint1.js (refer to Listing 2.5), you can see that just like a vertex shader, a fragment
shader is executed from main():

Draw a Point (Version 1)

35

9 // Fragment shader program
10 var FSHADER_SOURCE =
11
'void main() {\n' +
12
' gl_FragColor = vec4(1.0, 0.0, 0.0, 1.0);\n' +
13
'}\n';

The job of the shader is to set the color of the point as its per-fragment operation, which
is carried out at line 12. gl_FragColor is a built-in variable only available in a fragment
shader; it controls the color of a fragment, as shown in Table 2.4.
Table 2.4

The Built-In Value Available in a Fragment Shader

Type and Variable Name

Description

vec4 gl_FragColor

Specify the color of a fragment (in RGBA)

When we assign a color value to the built-in variable, the fragment is displayed using
that color. Just like the position in the vertex shader, the color value is a vec4 data type
consisting of four floating point numbers representing the RGBA values. In this sample
program, a red point will be displayed because you assign (1.0, 0.0, 0.0, 1.0) to the
variable.

The Draw Operation
Once you set up the shaders, the remaining task is to draw the shape, or in our case, a
point. As before, you need to clear the drawing area in a similar way to that described
in HelloCanvas.js. Once the drawing area is cleared, you can draw the point using gl.
drawArrays(), as in line 39:
39 gl.drawArrays(gl.POINTS, 0, 1);
gl.drawArrays() is a powerful function that is capable of drawing a variety of basic
shapes, as detailed in the following box.

36

CHAPTER 2

Your First Step with WebGL

gl.drawArrays(mode, first, count)
Execute a vertex shader to draw shapes specified by the mode parameter.
Parameters

mode

Specifies the type of shape to be drawn. The following symbolic
constants are accepted: gl.POINTS, gl.LINES, gl.LINE_STRIP,
gl.LINE_LOOP, gl.TRIANGLES, gl.TRIANGLE_STRIP, and gl.
TRIANGLE_FAN.

first

Specifies which vertex to start drawing from (integer).

count

Specifies the number of vertices to be used (integer).

Return value

None

Errors

INVALID_ENUM mode is none of the preceding values.
INVALID_VALUE first is negative or count is negative.

In this sample program, because you are drawing a point, you specify gl.POINTS as the
mode in the first parameter. The second parameter is set to 0 because you are starting from
the first vertex. The third parameter, count, is 1 because you are only drawing 1 point in
this sample program.
Now, when the program makes a call to gl.drawArrays(), the vertex shader is executed
count times, each time working with the next vertex. In this sample program, the shader
is executed once (count is set to 1) because we only have one vertex: our point. When
the shader is executed, the function main() in the shader is called, and then each line
in the function is executed sequentially, resulting in (0.0, 0.0, 0.0, 1.0) being assigned to
gl_Position (line 5) and then 10.0 assigned to gl_PointSize (line 6).
Once the vertex shader executes, the fragment shader is executed by calling its main()
function which, in this example, assigns the color value (red) to gl_FragColor (line 12).
As a result, a red point of 10 pixels is drawn at (0.0, 0.0, 0.0, 1.0), or the center of the
drawing area (see Figure 2.15).

Draw a Point (Version 1)

37

Vertex Shader
function main() {
var gl=getWebGL…
…
i nitShaders ( … ) ;
…
g l. d r a w A r r a y s ( … ) ;
}

void main() {

gl_Position

JavaScript

Fragment
Shader
void main() {

gl_Position =
gl_PointSize = 10.0;
}

gl_FragColor =

gl_PointSize

vec4(0.0, 0.0, 0.0, 1.0);

Color Buffer

gl_FragColor

Per-fragment operation

Per-vertex operation

vec4(1.0, 0.0, 0.0, 1.0);
}

W e bG L S yste m
Figure 2.15

The behavior of shaders

At this stage, you should have a rough understanding of the role of a vertex shader and a
fragment shader and how they work. In the rest of this chapter, you’ll build on this basic
understanding through a series of examples, allowing you to become more accustomed to
WebGL and shaders. However, before that, let’s quickly look at how WebGL describes the
position of shapes using its coordinate system.

The WebGL Coordinate System
Because WebGL deals with 3D graphics, it uses a three-dimensional coordinate system
along the x-, y-, and z-axis. This coordinate system is easy to understand because our
world has the same three dimensions: width, height, and depth. In any coordinate system,
the direction of each axis is important. Generally, in WebGL, when you face the computer
screen, the horizontal direction is the x-axis (right direction is positive), the vertical direction is the y-axis (up direction is positive), and the direction from the screen to the viewer
is the z-axis (the left side of Figure 2.16). The viewer’s eye is located at the origin (0.0, 0.0,
0.0), and the line of sight travels along the negative direction of the z-axis, or from you
into the screen (see the right side of Figure 2.16). This coordinate system is also called the
right-handed coordinate system because it can be expressed using the right hand (see
Figure 2.17) and is the one normally associated with WebGL. Throughout this book, we’ll
use the right-handed coordinate system as the default for WebGL. However, you should
note that it’s actually more complex than this. In fact, WebGL is neither left handed nor
right handed. This is explained in detail in Appendix D, “WebGL/OpenGL: Left or Right
Handed?,” but it’s safe to treat WebGL as right handed for now.
y

y
S cre e n

Line of sight
x

z

Figure 2.16
38

z

x

WebGL coordinate system

CHAPTER 2

Your First Step with WebGL

y

x

z

Figure 2.17

The right-handed coordinate system

As you have already seen, the drawing area specified for a  element in JavaScript
is different from WebGL’s coordinate system, so a mapping is needed between the two.
By default, as you see in Figure 2.18, WebGL maps the coordinate system to the area as
follows:

• The center position of a : (0.0, 0.0, 0.0)
• The two edges of the x-axis of the : (–1.0, 0.0, 0.0) and (1.0, 0.0, 0.0)
• The two edges of the y-axis of the : (0.0, –1.0, 0.0) and (0.0, 1.0, 0.0)

y

drawing area
(0.0, 0.0, 0.0)

x
(-1.0, 0.0, 0.0)

(1.0, 0.0, 0.0)

(0.0, -1.0, 0.0)

Figure 2.18

The  drawing area and WebGL coordinate system

As previously discussed, this is the default. It’s possible to use another coordinate
system, which we’ll discuss later, but for now this default coordinate system will be used.
Additionally, to help you stay focused on the core functionality of WebGL, the example
programs will mainly use the x and y coordinates and not use the z or depth coordinate.
Therefore, until Chapter 7, the z-axis value will generally be specified as 0.0.

Draw a Point (Version 1)

39

Experimenting with the Sample Program
First, you can modify line 5 to change the position of the point and gain a better understanding of the WebGL coordinate system. For example, let’s change the x coordinate
from 0.0 to 0.5 as follows:
5

'

gl_Position = vec4(0.5, 0.0, 0.0, 1.0);\n' +

Save the modified HelloPoint1.js and click the Reload button on your browser to reload
it. You will see that the point has moved and is now displayed on the right side of the
 area (see Figure 2.19, left side).
Now change the y coordinate to move the point toward the top of the  as
follows:
5

'

gl_Position = vec4(0.0, 0.5, 0.0, 1.0);\n' +

Again, save the modified HelloPoint1.js and reload it. This time, you can see the point
has moved and is displayed in the upper part of the canvas (see Figure 2.19, right side).

Figure 2.19

Modifying the position of the point

As another experiment, let’s try changing the color of the point from red to green by
modifying line 12, as follows:
12

'

gl_FragColor = vec4(0.0, 1.0, 0.0, 1.0);\n' +

Let’s conclude this section with a quick recap. You’ve been introduced to the two basic
shaders we use in WebGL—the vertex shader and the fragment shader—and seen how,
although they use their own language, they can be executed from within JavaScript.
You’ve also seen that the basic processing flow of a WebGL application using shaders is
the same as in other types of WebGL applications. A key lesson from this section is that
a WebGL program consists of a JavaScript program executing in conjunction with shader
programs.
40

CHAPTER 2

Your First Step with WebGL

For those of you with experience in using OpenGL, you may feel that something is
missing; there is no code to swap color buffers. One of the significant features of WebGL
is that it does not need to do that. For more information, see Appendix A, “No Need to
Swap Buffers in WebGL.”

Draw a Point (Version 2)
In the previous section, you explored drawing a point and the related core functions of
shaders. Now that you understand the fundamental behavior of a WebGL program, let’s
examine how to pass data between JavaScript and the shaders. HelloPoint1 always draws
a point at the same position because its position is directly written (“hard-coded”) in the
vertex shader. This makes the example easy to understand, but it lacks flexibility. In this
section, you’ll see how a WebGL program can pass a vertex position from JavaScript to
the vertex shader and then draw a point at that position. The name of the program is
HelloPoint2, and although the result of the program is the same as HelloPoint1, it’s a
flexible technique you will use in future examples.

Using Attribute Variables
Our goal is to pass a position from the JavaScript program to the vertex shader. There
are two ways to pass data to a vertex shader: attribute variable and uniform variable (see
Figure 2.20). The one you use depends on the nature of the data. The attribute variable
passes data that differs for each vertex, whereas the uniform variable passes data that is
the same (or uniform) in each vertex. In this program, you will use the attribute variable
because each vertex generally has different coordinates.
JavaScript
Color Buffer
attribute variable

u n i f o r m v ar i a b l e

Figure 2.20

gl_FragColor

Fragment
Shader

gl_PointSize

Vertex Shader

gl_Position

WebGL System
function main() {
var gl=getWebGL…
…
initShaders(…);
…
}

Two ways to pass data to a vertex shader

The attribute variable is a GLSL ES variable which is used to pass data from the world
outside a vertex shader into the shader and is only available to vertex shaders.

Draw a Point (Version 2)

41

To use the attribute variable, the sample program involves the following three steps:
1. Prepare the attribute variable for the vertex position in the vertex shader.
2. Assign the attribute variable to the gl_Position variable.
3. Pass the data to the attribute variable.
Let’s look at the sample program in more detail to see how to carry out these steps.

Sample Program (HelloPoint2.js)
In HelloPoint2 (see Listing 2.6), you draw a point at a position the JavaScript program
specifies.
Listing 2.6
1
2
3
4
5
6
7
8
9
10

HelloPoint2.js

// HelloPoint2.js
// Vertex shader program
var VSHADER_SOURCE =
'attribute vec4 a_Position;\n' +
'void main() {\n' +
' gl_Position = a_Position;\n' +
' gl_PointSize = 10.0;\n' +
'}\n';
// Fragment shader program
... snipped because it is the same as HelloPoint1.js

15
16 function main() {
17
// Retrieve  element
18
var canvas = document.getElementById('webgl');
19
20
// Get the rendering context for WebGL
21
var gl = getWebGLContext(canvas);
...
26
27
// Initialize shaders
28
if (!initShaders(gl, VSHADER_SOURCE, FSHADER_SOURCE)) {
...
31
}
32
33
// Get the storage location of attribute variable
34
var a_Position = gl.getAttribLocation(gl.program, 'a_Position');
35
if (a_Position < 0) {
36
console.log('Failed to get the storage location of a_Position');
37
return;

42

CHAPTER 2

Your First Step with WebGL

38
39
40
41
42
43
44
45
46
47
48
49
50
51

}
// Pass vertex position to attribute variable
gl.vertexAttrib3f(a_Position, 0.0, 0.0, 0.0);
// Set the color for clearing 
gl.clearColor(0.0, 0.0, 0.0, 1.0);
// Clear 
gl.clear(gl.COLOR_BUFFER_BIT);
// Draw a point
gl.drawArrays(gl.POINTS, 0, 1);
}

As you can see, the attribute variable is prepared within the shader on line 4:
4

'attribute vec4 a_Position;\n' +

In this line, the keyword attribute is called a storage qualifier, and it indicates that the
following variable (in this case, a_Position) is an attribute variable. This variable must
be declared as a global variable because data is passed to it from outside the shader. The
variable must be declared following a standard pattern  
, as shown in Figure 2.21.
Storage Qualifier

Type

Variable Name

attribute vec4 a_Position;
Figure 2.21

The declaration of the attribute variable

In line 4, you declare a_Position as an attribute variable with data type vec4 because,
as you saw in Table 2.2, it will be assigned to gl_Position, which always requires a vec4
type.
Note that throughout this book, we have adopted a programming convention in which
all attribute variables have the prefix a_, and all uniform variables have the prefix u_ to
easily determine the type of variables from their names. Obviously, you can use your own
convention when writing your own programs, but we find this one simple and clear.
Once a_Position is declared, it is assigned to gl_Position at line 6:
6

'

gl_Position = a_Position;\n' +

At this point, you have completed the preparation in the shader for receiving data from
the outside. The next step is to pass the data to the attribute variable from the JavaScript
program.
Draw a Point (Version 2)

43

Getting the Storage Location of an Attribute Variable
As you saw previously, the vertex shader program is set up in the WebGL system using
the convenience function initShaders(). When the vertex shader is passed to the
WebGL system, the system parses the shader, recognizes it has an attribute variable, and
then prepares the location of its attribute variable so that it can store data values when
required. When you want to pass data to a_Position in the vertex shader, you need to
ask the WebGL system to give you the location it has prepared, which can be done using
gl.getAttribLocation(), as shown in line 34:
33
34
35
36
37
38

// Get the location of attribute variable
var a_Position = gl.getAttribLocation(gl.program, 'a_Position');
if (a_Position < 0) {
console.log('Fail to get the storage location of a_Position');
return;
}

The first argument of this method specifies a program object that holds the vertex shader
and the fragment shader. You will examine the program object in Chapter 8, but for
now, you can just specify gl.program as the argument here. Note that you should use
gl.program only after initShaders() has been called because initShaders() assigns the
program object to the variable. The second parameter specifies the attribute variable name
(in this case a_Position) whose location you want to know.
The return value of this method is the storage location of the specified attribute variable.
This location is then stored in the JavaScript variable, a_Position, at line 34 for later use.
Again, for ease of understanding, this book uses JavaScript variable names for attribute
variables, which are the same as the GLSL ES attribute variable name. You can, of course,
use any variable name.
The specification of gl.getAttribLocation() is as follows:

gl.getAttribLocation(program, name)
Retrieve the storage location of the attribute variable specified by the name parameter.
Parameters

Return value

44

program

Specifies the program object that holds a vertex
shader and a fragment shader.

name

Specifies the name of the attribute variable
whose location is to be retrieved.

greater than or equal to 0

The location of the specified attribute variable.

-1

The specified attribute variable does not exist or
its name starts with the reserved prefix gl_ or
webgl_.

CHAPTER 2

Your First Step with WebGL

Errors

INVALID_OPERATION

program has not been successfully linked (See
Chapter 9.)

INVALID_VALUE

The length of name is more than the maximum
length (256 by default) of an attribute variable
name.

Assigning a Value to an Attribute Variable
Once you have the attribute variable location, you need to set the value using the
a_Position variable. This is performed at line 41 using the gl.vertexAttrib3f() method.
40
41

// Set vertex position to attribute variable
gl.vertexAttrib3f(a_Position, 0.0, 0.0, 0.0);

The following is the specification of gl.vertexAtrrib3f().

gl.vertexAttrib3f(location, v0, v1, v2)
Assign the data (v0, v1, and v2) to the attribute variable specified by location.
Parameters

location

Specifies the storage location of an attribute variable to be modified.

v0

Specifies the value to be used as the first element for the attribute
variable.

v1

Specifies the value to be used as the second element for the attribute variable.

v2

Specifies the value to be used as the third element for the attribute
variable.

Return value

None

Errors

INVALID_OPERATION

There is no current program object.

INVALID_VALUE

location is greater than or equal to the maximum
number of attribute variables (8, by default).

The first argument of the method call specifies the location returned by gl.getAttribLocation() at line 34. The second, third, and fourth arguments specify the floating point
number to be passed to a_Position representing the x, y, and z coordinates of the point.
After calling the method, these three values are passed as a group to a_Position, which
was prepared at line 4 in the vertex shader. Figure 2.22 shows the processing flow of
getting the location of the attribute variable and then writing a value to it.

Draw a Point (Version 2)

45

a_Position

a _ P o s it io n =
g l.getAttributeLocation( …, ‘ a _Position’);
…
g l. v e r t e x A t t r ib u t e 3 f ( a _ P o s it io n , 0 . 0 , 0 . 0 , 0 . 0 ) ;
…

Vertex Shader
v o id m a in ( ) {
g l _ P o s i t io n = a _ P o s i t io n ;
g l _ P o in t S iz e = 1 0 . 0 ;
}

gl_PointSize

f u n c t io n m a in ( ) {
v a r g l = g e tW e b G L C o n te x t ( ) ;
…
initShaders(gl, VSHADER_SOURCE, …) ;

gl_Position

JavaScript

W e bG L S yste m

}

a_Position

a _ P o s it io n =
g l.getAttributeLocation( …, ‘ a _Position’);
…
g l. v e r t e x A t t r ib u t e 3 f ( a _ P o s it io n , 0 . 0 , 0 . 0 , 0 . 0 ) ;
…

Vertex Shader
v o id m a in ( ) {
g l _ P o s i t io n = a _ P o s i t i o n ;
g l _ P o in t S iz e = 1 0 . 0 ;
}

gl_PointSize

f u n c t io n m a in ( ) {
v a r g l = g e tW e b G L C o n te x t ( ) ;
…
initShaders(gl, VSHADER_SOURCE, …) ;

gl_Position

JavaScript

W e bG L S yste m

}

Figure 2.22
the variable

Getting the storage location of an attribute variable and then writing a value to

a_Position is then assigned to gl_Position at line 6 in the vertex shader, in effect passing

the x, y, and z coordinates from your JavaScript, via the attribute variable into the shader,
where it’s written to gl_Position. So the program has the same effect as HelloPoint1,
where gl_Position is used as the position of a point. However, gl_Position has now been
set dynamically from JavaScript rather than statically in the vertex shader:
4
5
6
7
8

'attribute vec4 a_Position;\n' +
'void main() {\n' +
' gl_Position = a_Position;\n' +
' gl_PointSize = 10.0;\n' +
'}\n';

Finally, you clear the  using gl.clear() (line 47) and draw the point using
gl.drawArrays() (line 50) in the same way as in HelloPoint1.js.

46

CHAPTER 2

Your First Step with WebGL

As a final note, you can see a_Position is prepared as vec4 at line 4 in the vertex shader.
However, gl.vertexAttrib3f() at line 41 specifies only three values (x, y, and z), not
four. Although you may think that one value is missing, this method automatically
supplies the value 1.0 as the fourth value (see Figure 2.23). As you saw earlier, a default
fourth value of 1.0 for a color ensures it is fully opaque, and a default fourth value of 1.0
for a homogeneous coordinate maps a 3D coordinate into a homogenous coordinate, so
essentially the method is supplying a “safe” fourth value.

0.0 0.0 0.0 1.0

a_Position

V e rte x S h a d e r

fu n c tion m ain() {
var g l= getWebGLContext();
…
initShaders(gl, VS HADER_SOURCE, …);

g l_ P o s itio n = a _ P o s itio n ;
g l _ P o i n t S i z e = 1 0 .0 ;

Figure 2.23

}

0.0 0.0 0.0

gl_PointSize

v o id m a in ( ) {

a_Position =
gl.getAttributeLocation(…, ‘ a_Position’);
…
gl.vertexAttribute3f(a_Position, 0.0, 0.0, 0.0);
…
}

gl_Position

JavaScript

W ebGL System

The missing data is automatically supplied

Family Methods of gl.vertexAttrib3f()
gl.vertexAttrib3f() is part of a family of methods that allow you to set some or all of
the components of the attribute variable. gl.vertexAttrib1f() is used to assign a single
value (v0), glvertexAttrib2f() assigns two values (v0 and v1), and gl.vertexAttrib4f()

assigns four values (v0, v1, v2, and v3).

gl.vertexAttrib1f(location,
gl.vertexAttrib2f(location,
gl.vertexAttrib3f(location,
gl.vertexAttrib4f(location,

v0)
v0, v1)
v0, v1, v2)
v0, v1, v2, v3)

Assign data to the attribute variable specified by location. gl.vertexAttrib1f() indicates
that only one value is passed, and it will be used to modify the first component of the
attribute variable. The second and third components will be set to 0.0, and the fourth
component will be set to 1.0. Similarly, gl.vertexAttrib2f() indicates that values are
provided for the first two components, the third component will be set to 0.0, and the
fourth component will be set to 1.0. gl.vertexAttrib3f() indicates that values are
provided for the first three components, and the fourth component will be set to 1.0,
whereas gl.vertexAttrib4f() indicates that values are provided for all four components.

Draw a Point (Version 2)

47

Parameters

location

Specifies the storage location of the attribute variable.

v0, v1, v2, v3

Specifies the values to be assigned to the first, second,
third, and fourth components of the attribute variable.

Return value

None

Errors

INVALID_VALUE

location is greater than or equal to the maximum number
of attribute variables (8 by default).

The vector versions of these methods are also available. Their name contains “v” (vector),
and they take a typed array (see Chapter 4) as a parameter. The number in the method
name indicates the number of elements in the array. For example,
var position = new Float32Array([1.0, 2.0, 3.0, 1.0]);
gl.vertexAttrib4fv(a_Position, position);

In this case, 4 in the method name indicates that the length of the array is 4.

The Naming Rules for WebGL-Related Methods
You may be wondering what 3f in gl.vertexAttrib3f() actually means. WebGL bases
its method names on the function names in OpenGL ES 2.0, which as you now know,
is the base specification of WebGL. The function names in OpenGL comprise the three
components:   < parameter type>.
The name of each WebGL method also uses the same components: 
  as shown in Figure 2.24.

gl.vertexAttrib3f(location, v0, v1, v2 )
parameter type :
“ f ” i n d ic a t e s f l o a t i n g p o i n t n u m ber s a n d
“ i ” indicate s intege r numbers
numbe r of parameters

base method na me

Figure 2.24

The naming rules of WebGL-related methods

As you can see in the example, in the case of gl.vertexAttrib3f(), the base method
name is vertexAttrib, the number of parameters is 3, and the parameter type is f (that
is, float or floating point number). This method is a WebGL version of the function
glVertexAttrib3f() in OpenGL. Another character for the parameter type is i, which
indicates integer. You can use the following notation to represent all methods from gl.
vertexAttrib1f() to gl.vertexAttrib4f(): gl.vertexAttrib[1234]f().
Where [] indicates that one of the numbers in it can be selected.

48

CHAPTER 2

Your First Step with WebGL

When v is appended to the name, the methods take an array as a parameter. In this case,
the number in the method name indicates the number of elements in the array.
var positions = new Float32Array([1.0, 2.0, 3.0, 1.0]);
gl.vertexAttrib4fv(a_Position, positions);

Experimenting with the Sample Program
Now that you have the program to pass the position of a point from a JavaScript program
to a vertex shader, let’s change that position. For example, if you wanted to display a
point at (0.5, 0.0, 0.0), you would modify the program as follows:
33

gl.vertexAttrib3f(a_Position, 0.5, 0.0, 0.0);

You could use other family methods of gl.vertexAttrib3f() to perform the same task in
the following way:
gl.vertexAttrib1f(a_Position, 0.5);
gl.vertexAttrib2f(a_Position, 0.5, 0.0);
gl.vertexAttrib4f(a_Position, 0.5, 0.0, 0.0, 1.0);

Now that you are comfortable using attribute variables, let’s use the same approach to
change the size of the point from within your JavaScript program. In this case, you will
need a new attribute variable for passing the size to the vertex shader. Following the
naming rule used in this book, let’s use a_PointSize. As you saw in Table 2.2, the type of
gl_PointSize is float, so you need to prepare the variable using the same type as follows:
attribute float a_PointSize;

So, the vertex shader becomes:
2
3
4
5
6
7
8
9

// Vertex shader program
var VSHADER_SOURCE =
'attribute vec4 a_Position;\n' +
'attribute float a_PointSize; \n' +
'void main() {\n' +
' gl_Position = a_Position;\n' +
' gl_PointSize = a_PointSize;\n' +
'}\n';

Then, after you get the storage location of a_PointSize, to pass the size of the point to the
variable, you can use gl.vertexAttrib1f(). Because the type of a_PointSize is float, you
can use gl.vertexAttrib1f() as follows:
33
34

// Get the storage location of attribute variable
var a_Position = gl.getAttribLocation(gl.program, 'a_Position');
...

Draw a Point (Version 2)

49

39
40
41
42

var a_PointSize = gl.getAttribLocation(gl.program, 'a_PointSize');
// Set vertex position to attribute variable
gl.vertexAttrib3f(a_Position, 0.0, 0.0, 0.0);
gl.vertexAttrib1f(a_PointSize, 5.0);

At this stage, you might want to experiment a little with the program and make sure you
understand how to use these attribute variables and how they work.

Draw a Point with a Mouse Click
The previous program HelloPoint2 can pass the position of a point to a vertex shader
from a JavaScript program. However, the position is still hard-coded, so it is not so different from HelloPoint1, in which the position was also directly written in the shader.
In this section, you add a little more flexibility, exploiting the ability to pass the position
from a JavaScript program to a vertex shader, to draw a point at the position where the
mouse is clicked. Figure 2.25 shows the screen shot of ClickedPoint.4

Figure 2.25

ClickedPoint

This program uses an event handler to handle mouse-related events, which will be familiar to those of you who have written JavaScript programs.

Sample Program (ClickedPoints.js)
Listing 2.7 shows ClickedPoints.js. For brevity, we have removed code sections that are
the same as the previous example and replaced these with ....
4

© 2012 Marisuke Kunnya

50

CHAPTER 2

Your First Step with WebGL

Listing 2.7
1
2
3
4
5
6
7
8
9
10
16
17
18
19
20
21
27
28
31
32
33
34
40
41
47
48
49
50
51
52
53
54
55
56
57
58
59
60

ClickedPoints.js

// ClickedPoints.js
// Vertex shader program
var VSHADER_SOURCE =
'attribute vec4 a_Position;\n' +
'void main() {\n' +
' gl_Position = a_Position;\n' +
' gl_PointSize = 10.0;\n' +
'}\n';
// Fragment shader program
...
function main() {
// Retrieve  element
var canvas = document.getElementById('webgl');
// Get the rendering context for WebGL
var gl = getWebGLContext(canvas);
...
// Initialize shaders
if (!initShaders(gl, VSHADER_SOURCE, FSHADER_SOURCE)){
...
}
// Get the storage location of a_Position variable
var a_Position = gl.getAttribLocation(gl.program, 'a_Position');
...
// Register function (event handler) to be called on a mouse press
canvas.onmousedown = function(ev) { click(ev, gl, canvas, a_Position); };
...
gl.clear(gl.COLOR_BUFFER_BIT);
}
var g_points = []; // The array for a mouse press
function click(ev, gl, canvas, a_Position) {
var x = ev.clientX; // x coordinate of a mouse pointer
var y = ev.clientY; // y coordinate of a mouse pointer
var rect = ev.target.getBoundingClientRect();
x = ((x - rect.left) - canvas.height/2)/(canvas.height/2);
y = (canvas.width/2 - (y - rect.top))/(canvas.width/2);
// Store the coordinates to g_points array
g_points.push(x); g_points.push(y);

Draw a Point with a Mouse Click

51

61
62
63
64
65
66
67
68
69
70
71
72

// Clear 
gl.clear(gl.COLOR_BUFFER_BIT);
var len = g_points.length;
for(var i = 0; i < len; i+=2) {
// Pass the position of a point to a_Position variable
gl.vertexAttrib3f(a_Position, g_points[i], g_points[i+1], 0.0);
// Draw a point
gl.drawArrays(gl.POINTS, 0, 1);
}
}

Register Event Handlers
The processing flow from lines 17 to 39 is the same as HelloPoint2.js. These lines get the
WebGL context, initialize the shaders, and then retrieve the location of the attribute variable. The main differences from HelloPoint2.js are the registration of an event handler
(line 41) and the definition of the function click() as the handler (from line 51).
An event handler is an asynchronous callback function that handles input on a web page
from a user, such as mouse clicks or key presses. This allows you to create a dynamic web
page that can change the content that’s displayed according to the user’s input. To use an
event handler, you need to register the event handler (that is, tell the system what code to
run when the event occurs). The  supports special properties for registering event
handlers for a specific user input, which you will use here.
For example, when you want to handle mouse clicks, you can use the onmousedown property of the  to specify the event handler for a mouse click as follows:
40
41

// Register function (event handler) to be called on a mouse press
canvas.onmousedown = function(ev) { click(ev, gl, canvas, a_Position); };

Line 41 uses the statement function(){ ... } to register the handler:
function(ev){ click(ev, gl, canvas, a_Position); }

This mechanism is called an anonymous function, which, as its name suggests, is a
convenient mechanism when you define a function that does not need to have a name.
For those of you unfamiliar with this type of function, let’s explain the mechanism using
an example. Next, we define the variable thanks as follows using the form:
var thanks = function () { alert(' Thanks a million!'); }

We can execute the variable as a function as follows:
thanks(); // 'Thanks a million!' is displayed

52

CHAPTER 2

Your First Step with WebGL

You can see that the variable thanks can be operated as a function. These lines can be
rewritten as follows:
function thanks() { alert('Thanks a million!'); }
thanks(); // 'Thanks a million!' is displayed

So why do you need to use an anonymous function? Well, when you draw a point, you
need the following three variables: gl, canvas, and a_Position. These variables are local
variables that are prepared in the function main() in the JavaScript program. However,
when a mouse click occurs, the browser will automatically call the function that is registered to the ’s onmousedown property with a predefined single parameter (that is,
an event object that contains information about the mouse press. Therefore, usually, you
will register the event handler and define the function for it as follows:
canvas.onmousedown = mousedown; // Register "mousedown" as event handler
...
function mousedown(ev) { // Event handler: It takes one parameter "ev"
...
}

However, if you define the function in this way, it cannot access the local variables gl,
canvas, and a_Position to draw a point. The anonymous function at line 41 provides the
solution to make it possible to access them:
41

canvas.onmousedown = function(ev) { click(ev, gl, canvas, a_Position); };

In this code, when a mouse click occurs, the anonymous function function(ev) is called
first. Then it calls click(ev, gl, canvas, a_Position) with the local variables defined
in main(). Although this may seem a little complicated, it is actually quite flexible and
avoids the use of global variables, which is always good. Take a moment to make sure
you understand this approach, because you will often see this way of registering event
handlers in this book.

Handling Mouse Click Events
Let’s look at what the function click() is doing. The processing flow follows:
1. Retrieve the position of the mouse click and then store it in an array.
2. Clear .
3. For each position stored in the array, draw a point.
50
51
52
53
54

var g_points = []; // The array for mouse click positions
function click(ev, gl, canvas, a_Position) {
var x = ev.clientX; // x coordinate of a mouse pointer
var y = ev.clientY; // y coordinate of a mouse pointer
var rect = ev.target.getBoundingClientRect();

Draw a Point with a Mouse Click

53

55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72

x = ((x - rect.left) - canvas.height/2)/(canvas.height/2);
y = (canvas.width/2 - (y - rect.top))/(canvas.width/2);
// Store the coordinates to a_points array
g_points.push(x); g_points.push(y);
// Clear 
gl.clear(gl.COLOR_BUFFER_BIT);

<- (1)

<- (2)

var len = g_points.length;
for(var i = 0; i < len; i+=2) {
// Pass the position of a point to a_Position variable
gl.vertexAttrib3f(a_Position, g_points[i], g_points[i+1], 0.0);

<- (3)

// Draw a point
gl.drawArrays(gl.POINTS, 0, 1);
}
}

The information about the position of a mouse click is stored as an event object and
passed by the browser using the argument ev to the function click(). ev holds the position information, and you can get the coordinates by using ev.clientX and ev.clientY,
as shown in lines 52 and 53. However, you cannot use the coordinates directly in this
sample program for two reasons:
1. The coordinate is the position in the “client area” in the browser, not in the
 (see Figure 2.26).
The origin of client area
The origin 
The client area of a browser
The drawing area of 
(0, 0)

x
Click!

y

ev.clientY

e v . c lie n t X a n d e v . c lie n t Y
h o ld t h e p o s it io n in t h e
c lie n t a r e a

ev.clientX

Figure 2.26


The coordinate system of a browser’s client area and the position of the

2. The coordinate system of the  is different from that of WebGL (see Figure
2.27) in terms of their origin and the direction of the y-axis.
54

CHAPTER 2

Your First Step with WebGL

y

x

(0, 0)


drawing area

(0.0, 1.0, 0.0)

drawing area

(0.0, 0.0, 0.0)

x
(-1.0, 0.0, 0.0)

y

(0.0, -1.0, 0.0)

(400, 400)

Figure 2.27

(1.0, 0.0, 0.0)

The coordinate system of  (left) and that of WebGL on  (right)

First you need to transform the coordinates from the browser area to the canvas, and then
you need to transform them into coordinates that WebGL understands. Let’s look at how
to do that.
These transformations are performed at lines 56 and 57 in the sample program:
52
53
54
55
56
57

var
var
var
...
x =
y =

x = ev.clientX;
y = ev.clientY;
rect = ev.target.getBoundingClientRect();
((x - rect.left) - canvas.width/2)/(canvas.width/2);
(canvas.height/2 - (y - rect.top))/(canvas.height/2);

Line 54 gets the position of the  in the client area. The rect.left and rec.top
indicate the position of the origin of the  in the browser’s client area (refer to
Figure 2.26). So, (x – rect.left) at line 56 and (y – rect.top) at line 57 slide the position (x, y) in the client area to the correct position on the  element.
Next, you need to transform the  position into the WebGL coordinate system
shown in Figure 2.27. To perform the transformation, you need to know the center position of the . You can get the size of the  by canvas.height (in this case,
400) and canvas.width (in this case, 400). So, the center of the element will be (canvas.
height/2, canvas.width/2).
Next, you can implement this transformation by sliding the origin of the 
into the origin of the WebGL coordinate system located at the center of . The
((x - rect.left) - canvas.width/2) and (canvas.height/2 - (y -rect.top)) perform
the transformation.
Finally, as shown in Figure 2.27, the range of the x-axis in the  goes from 0 to
canvas.width (400), and that of the y-axis goes from 0 to canvas.height (400). Because
Draw a Point with a Mouse Click

55

the axes in WebGL range from –1.0 to 1.0, the last step is to map the range of the
 coordinate system to that of the WebGL coordinate system by dividing the x
coordinate by canvas.width/2 and the y coordinate by canvas.height/2. You can see this
division in lines 56 and 57.
Line 59 stores the resulting position of the mouse click in the array g_points using the
push() method, which appends the data to the end of the array:
59

g_points.push(x); g_points.push(y);

So every time a mouse click occurs, the position of the click is appended to the array, as
shown in Figure 2.28. (The length of the array is automatically stretched.) Note that the
index of an array starts from 0, so the first position is g_points[0].

the contents of g_points [ ]
x coordinate
of the 1st
clicked point

y coordinate
of the 1st
clicked point

x coordinate
of the 2nd
clicked point

y coordinate
of the 2nd
clicked point

x coordinate
of the 3rd
clicked point

y coordinate
of the 3rd
clicked point

g_points[0]

g_points[1]

g_points[2]

g_points[3]

g_points[4]

g_points[5]

Figure 2.28

…

The content of g_points

You may be wondering why you need to store all the positions rather than just the most
recent mouse click. This is because of the way WebGL uses the color buffer. You will
remember from Figure 2.10 that in the WebGL system, first the drawing operation is
performed to the color buffer, and then the system displays its content to the screen. After
that, the color buffer is reinitialized and its content is lost. (This is the default behavior,
which you’ll investigate in the next section.) Therefore, it is necessary to save all the positions of the clicked points in the array, so that on each mouse click, the program can
draw all the previous points as well as the latest. For example, the first point is drawn at
the first mouse click. The first and second points are drawn at the second mouse click. The
first, second, and third points are drawn on the third click, and so on.
Returning to the program, the  is cleared at line 62. After that, the for statement
at line 65 passes each position of the point stored in g_points to a_Position in the vertex
shader. Then gl.drawArrays() draws the point at that position:
65
66
67

for(var i = 0; i < len; i+=2) {
// Pass the position of a point to a_Position variable
gl.vertexAttrib3f(a_Position, g_points[i], g_points[i+1], 0.0);

Just like in HelloPoint2.js, you will use gl.vertexAttrib3f() to pass the point position
to the attribute variable a_Position, which was passed as the fourth parameter of click()
at line 51.

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The array g_points holds the x coordinate and y coordinate of the clicked points, as
shown in Figure 2.28. Therefore, g_points[i] holds the x coordinate, and g_points[i+1]
holds the y coordinate, so the loop index i in the for statement at line 65 is incremented
by 2 using the convenient + operator.
Now that you have completed the preparations for drawing the point, the remaining task
is just to draw it using gl.drawArrays():
69
70

// Draw a point
gl.drawArrays(gl.POINTS, 0, 1);

Although it’s a little complicated, you can see that the use of event handlers combined
with attribute variable provides a flexible and generic means for user input to change what
WebGL draws.

Experimenting with the Sample Program
Now let’s experiment a little with this sample program ClickedPoints. After loading
ClickedPoints.html in your browser, every time you click, it draws a point at the clicked
position, as shown in Figure 2.26.
Let’s examine what will happen when you stop clearing the  at line 62.
Comment out the line as follows, and then reload the modified file into your browser.
61
62
63
64
65
66
67
68
69
70
71
72

// Clear 
// gl.clear(gl.COLOR_BUFFER_BIT);
var len = g_points.length;
for(var i = 0; i < len; i+=2) {
// Pass the position of a point to a_Position variable
gl.vertexAttrib3f(a_Position, g_points[i], g_points[i+1], 0.0);
// Draw a point
gl.drawArrays(gl.POINTS, 0, 1);
}
}

After running the program, you initially see a black background, but at the first click,
you see a red point on a white background. This is because WebGL reinitializes the color
buffer to the default value (0.0, 0.0, 0.0, 0.0) after drawing the point (refer to Table 2.1).
The alpha component of the default value is 0.0, which means the color is transparent.
Therefore, the color of the  becomes transparent, so you see the background color
of the web page (white, in this case) through the  element. If you don’t want this
behavior, you should use gl.clearColor() to specify the clear color and then always call
gl.clear() before drawing something.

Draw a Point with a Mouse Click

57

Another interesting experiment is to look at how to simplify the code. In ClickedPoints.
js, the x and y coordinates are stored separately in the g_points array. However, you can
store the x and y coordinates as a group into the array as follows:
58
59

// Store the coordinates to a_points array
g_points.push([x, y]);

<- (1)

In this case, the new two-element array [x, y] is stored as an element in the array
g_points. Conveniently, JavaScript can store an array in an array.
You can retrieve the x and y coordinates from the array as follows. First, you retrieve an
element from the array by specifying its index (line 66). Because the element itself is an
array containing (x, y), you can retrieve the first element and the second element as the
x and y coordinate of each point from the array (line 67):
65
66
67
71

for(var i = 0; i < len; i++) {
var xy = g_points[i];
gl.vertexAttrib3f(a_Position, xy[0], xy[1], 0.0);
...
}

In this way, you can deal with the x and y coordinates as a group, simplifying your
program and increasing the readability.

Change the Point Color
By now you should have a good feel for how the shaders work and how to pass data to
them from your JavaScript program. Let’s build on your understanding by constructing a
more complex program that draws points whose colors vary depending on their position
on the .
You already studied how to change the color of a point when you looked at HelloPoint1.
In that example, you modified the fragment shader program directly to embed the color
value into the shader. In this section, let’s construct the program so that you can specify
the color of a point from JavaScript. This is similar to HelloPoint2 earlier, where you
passed the position of a point from a JavaScript program to the vertex shader. However,
in this sample program, you need to pass the data to a “fragment shader,” not to a vertex
shader.
The name of the sample program is ColoredPoints. If you load the example into your
browser, the result is the same as ClickedPoints except that each point’s color varies
depending on its position (see Figure 2.29).

58

CHAPTER 2

Your First Step with WebGL

Figure 2.29

ColoredPoints

To pass data to a fragment shader, you can use a uniform variable and follow the same
steps that you used when working with attribute variables. However, this time the target is
a fragment shader, not a vertex shader:
1. Prepare the uniform variable for the color in the fragment shader.
2. Assign the uniform variable to the gl_FragColor variable.
3. Pass the color data to the uniform variable from the JavaScript program.
Let’s look at the sample program and see how these steps are programmed.

Sample Program (ColoredPoints.js)
The vertex shader of this sample program is the same as in ClickedPoints.js. However,
this time, the fragment shader plays a more important role because the program changes
the color of the point dynamically and, as you will remember, fragment shaders handle
colors. Listing 2.8 shows ColoredPoints.js.
Listing 2.8

ColoredPoints.js

1 // ColoredPoints.js
2 // Vertex shader program
3 var VSHADER_SOURCE =
4
'attribute vec4 a_Position;\n' +
5
'void main() {\n' +
6
' gl_Position = a_Position;\n' +
7
' gl_PointSize = 10.0;\n' +

Change the Point Color

59

8
'}\n';
9
10 // Fragment shader program
11 var FSHADER_SOURCE =
12
'precision mediump float;\n' +
13
'uniform vec4 u_FragColor;\n' + // uniform variable
<- (1)
14
'void main() {\n' +
15
' gl_FragColor = u_FragColor;\n' +
<- (2)
16
'}\n';
17
18 function main() {
...
29
// Initialize shaders
30
if (!initShaders(gl, VSHADER_SOURCE, FSHADER_SOURCE)) {
...
33
}
34
35
// Get the storage location of a_Position variable
36
var a_Position = gl.getAttribLocation(gl.program, 'a_Position');
...
42
// Get the storage location of u_FragColor variable
43
var u_FragColor = gl.getUniformLocation(gl.program, 'u_FragColor');
...
49
// Register function (event handler) to be called on a mouse press
50
canvas.onmousedown = function(ev){ click(ev, gl, canvas, a_Position,
➥u_FragColor) };
...
56
gl.clear(gl.COLOR_BUFFER_BIT);
57 }
58
59 var g_points = []; // The array for a mouse press
60 var g_colors = []; // The array to store the color of a point
61 function click(ev, gl, canvas, a_Position, u_FragColor) {
62
var x = ev.clientX; // x coordinate of a mouse pointer
63
var y = ev.clientY; // y coordinate of a mouse pointer
64
var rect = ev.target.getBoundingClientRect();
65
66
x = ((x - rect.left) - canvas.width/2)/(canvas.width/2);
67
y = (canvas.height/2 - (y - rect.top))/(canvas.height/2);
68
69
// Store the coordinates to g_points array
70
g_points.push([x, y]);
71
// Store the color to g_colors array
72
73

60

if(x >= 0.0 && y >= 0.0) {
// First quadrant
g_colors.push([1.0, 0.0, 0.0, 1.0]); // Red

CHAPTER 2

Your First Step with WebGL

74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95 }

} else if(x < 0.0 && y < 0.0) {
// Third quadrant
g_colors.push([0.0, 1.0, 0.0, 1.0]); // Green
} else {
// Others
g_colors.push([1.0, 1.0, 1.0, 1.0]); // White
}
// Clear 
gl.clear(gl.COLOR_BUFFER_BIT);
var len = g_points.length;
for(var i = 0; i < len; i++) {
var xy = g_points[i];
var rgba = g_colors[i];
// Pass the position of a point to a_Position variable
gl.vertexAttrib3f(a_Position, xy[0], xy[1], 0.0);
// Pass the color of a point to u_FragColor variable
gl.uniform4f(u_FragColor, rgba[0],rgba[1],rgba[2],rgba[3]);
// Draw a point
gl.drawArrays(gl.POINTS, 0, 1);

<-(3)

}

Uniform Variables
You likely remember how to use an attribute variable to pass data from a JavaScript
program to a vertex shader. Unfortunately, the attribute variable is only available in a
vertex shader, and when using a fragment shader, you need to use a uniform variable.
There is an alternative mechanism, a varying variable (bottom of Figure 2.30); however,
it is more complex, so you won’t use it until Chapter 5.
JavaScript
Color Buffer

varying variable

Figure 2.30

gl_Position

Vertex Shader

Fragment
Shader

gl_PointSize

function main() {
var gl=getWebGL…
…
i nitShaders ( … ) ;
…
}

gl_FragColor

uni fo rm v ar i abl e

attribu te variable

varying variable

Two ways of passing a data to a fragment shader
Change the Point Color

61

When you were introduced to attribute variables, you saw that a uniform variable is a
variable for passing “uniform” (nonvariable) data from a JavaScript program to all vertices
or fragments in the shader. Let’s now use that property.
Before using a uniform variable, you need to declare the variable using the form    in the same way (see Figure 2.31) as declaring an
attribute variable. (See the section “Sample Program (HelloPoint2.js).”)5

Storage Qualifier

Type

Variable Name

uniform vec4 u_FragColor;
Figure 2.31

The declaration of the uniform variable

In this sample program, the uniform variable u_FragColor is named after the variable
gl_FragColor because you will assign the uniform variable to the gl_FragColor later.
The u_ in u_FragColor is part of our programming convention and indicates that it is a
uniform variable. You need to specify the same data type as gl_FragColor for u_FragColor
because you can only assign the same type of variable to gl_FragColor. Therefore, you
prepare u_FragColor at line 13, as follows:
10
11
12
13
14
15
16

// Fragment shader program
var FSHADER_SOURCE =
'precision mediump float;\n' +
'uniform vec4 u_FragColor;\n' + // uniform variable
'void main() {\n' +
' gl_FragColor = u_FragColor;\n' +
'}\n';

Note that line 12 specifies the range and precision of variables by using a precision qualifier, in this case medium precision, which is covered in detail in Chapter 5.
Line 15 assigns the color in the uniform variable u_FragColor to gl_FragColor, which
causes the point to be drawn in whatever color is passed. Passing the color through the
uniform variable is similar to using an attribute variable; you need to retrieve the storage
location of the variable and then write the data to the location within the JavaScript
program.

Retrieving the Storage Location of a Uniform Variable
You can use the following method to retrieve the storage location of a uniform variable.

5

In GLSL ES, you can only specify float data types for an attribute variable; however, you can
specify any type for a uniform variable. (See Chapter 6 for more details.)

62

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gl.getUniformLocation(program, name)
Retrieve the storage location of the uniform variable specified by the name parameter.
Parameters

Return value

Errors

program

Specifies the program object that holds a vertex
shader and a fragment shader.

name

Specifies the name of the uniform variable whose location is to be retrieved.

non-null

The location of the specified uniform variable.

null

The specified uniform variable does not exist or its
name starts with the reserved prefix gl_ or webgl_.

INVALID_OPERATION

program has not been successfully linked
(See Chapter 9.)

INVALID_VALUE

The length of name is more than the maximum length
(256 by default) of a uniform variable.

The functionality and parameters of this method are the same as in gl.getAttribLocation(). However, the return value of this method is null, not –1, if the uniform variable
does not exist or its name starts with a reserved prefix. For this reason, unlike attribute
variables, you need to check whether the return value is null. You can see this error
checking line 44. In JavaScript, null can be treated as false when checking the condition
of an if statement, so you can use the ! operator to check the result:
42
43
44
45
46
47

// Get the storage location of uniform variable
var u_FragColor = gl.getUniformLocation(gl.program, 'u_FragColor');
if (!u_FragColor) {
console.log('Failed to get u_FragColor variable');
return;
}

Assigning a Value to a Uniform Variable
Once you have the location of the uniform variable and the WebGL method, gl.
uniform4f() is used to write data to it. It has the same functionality and parameters as
those of gl.vertexAttrib[1234]f().

Change the Point Color

63

gl.uniform4f(location, v0, v1, v2, v3)
Assign the data specified by v0, v1, v2, and v3 to the uniform variable specified by
location.
Parameters

location

Specifies the storage location of a uniform variable to
be modified.

v0

Specifies the value to be used as the first element of
the uniform variable.

v1

Specifies the value to be used as the second element
of the uniform variable.

v2

Specifies the value to be used as the third element of
the uniform variable.

v3

Specifies the value to be used as the fourth element of
the uniform variable.

Return value

None

Errors

INVALID_OPERATION

There is no current program object.
location is an invalid uniform variable location.

Let’s look at the portion of the sample program where the gl.uniform4f() method is used
to assign the data (line 91). As you can see, several processing steps are needed in advance:
71
72
73
74
75
76
77
78
83
84
85
86
91

// Store the color to g_colors array
if(x >= 0.0 && y >= 0.0) {
// First quadrant
g_colors.push([1.0, 0.0, 0.0, 1.0]); // Red
} else if(x < 0.0 && y < 0.0) {
// Third quadrant
g_colors.push([0.0, 1.0, 0.0, 1.0]); // Green
} else {
// Other quadrants
g_colors.push([1.0, 1.0, 1.0, 1.0]); // White
}
...
var len = g_points.length;
for(var i = 0; i < len; i++) {
var xy = g_points[i];
var rgba = g_colors[i];
...
gl.uniform4f(u_FragColor, rgba[0], rgba[1], rgba[2], rgba[3]);

To understand this program logic, let’s return to the goal of this program, which is to vary
the color of a point according to where on the  the mouse is clicked. In the first

64

CHAPTER 2

Your First Step with WebGL

quadrant, the color is set to red; in the third quadrant, the color is green; in the other
quadrants, the color is white (see Figure 2.32).

y
S e c o n d q u ad r a nt
(white)

First quadrant
(red)

x
F o u rt h q u a d r a n t
(white)

Third quadrant
(green)

Figure 2.32

The name of each quadrant in a coordinate system and its drawing colors

The code from lines 72 to 78 simply determines which quadrant the mouse click was in
and then, based on that, writes the appropriate color into the array g_colors. Finally, at
line 84, the program loops through the points, passing the correct color to the uniform
variable u_FragColor at line 91. This causes WebGL to write all the points stored so far to
the color buffer, which then results in the browser display being updated.
Before finishing this chapter, let’s take a quick look at the rest of the family methods for
gl.uniform[1234]f().

Family Methods of gl.uniform4f()
gl.uniform4f() also has a family of methods. gl.uniform1f() is a method to assign a
single value (v0), gl.uniform2f() assigns two values (v0 and v1), and gl.uniform3f()
assigns three values (v0, v1, and v2).

gl.uniform1f(location, v0)
gl.uniform2f(location, v0, v1)

gl.uniform3f(location, v0, v1, v2)
gl.uniform4f(location, v0, v1, v2, v3)
Assign data to the uniform variable specified by location. gl.uniform1f() indicates
that only one value is passed, and it will be used to modify the first component of
the uniform variable. The second and third components will be set to 0.0, and the
fourth component will be set to 1.0. Similarly, gl.uniform2f() indicates that values are
provided for the first two components, the third component will be set to 0.0, and the
fourth component will be set to 1.0. gl.uniform3f() indicates that values are provided
for the first three components and the fourth component will be set to 1.0, whereas
gl.unifrom4f() indicates that values are provided for all four components.

Change the Point Color

65

Parameters

location

Specifies the storage location of a uniform variable.

v0, v1, v2, v3

Specifies the values to be assigned to the first,
second, third, and fourth component of the uniform
variable.

Return value

None

Errors

INVALID_OPERATION

There is no current program object.
location is an invalid uniform variable location.

Summary
In this chapter, you saw the core functions of WebGL and how to use them. In particular, you learned about shaders, which are the main mechanism used in WebGL to draw
graphics. Based on this, you constructed several sample programs starting with a simple
program to draw a red point, and then you added complexity by changing its position
based on a mouse click and changed its color. In both cases, these examples helped you
understand how to pass data from a JavaScript program to the shaders, which will be critical for future examples.
The shapes found in this chapter were still just two-dimensional points. However, you can
apply the same knowledge of the use of the core WebGL functions and shaders to more
complex shapes and to three-dimensional objects.
A key learning point in this chapter is this: A vertex shader performs per-vertex operations, and a fragment shader performs per-fragment operations. The following chapters
explain some of the other functions of WebGL, while slowly increasing the complexity of
the shapes that you manipulate and display on the screen.

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Chapter 3

Drawing and Transforming Triangles

Chapter 2, “Your First Step with WebGL,” explained the basic approach to drawing WebGL
graphics. You saw how to retrieve the WebGL context and clear a  in preparation for
drawing your 2D/3DCG. You then explored the roles and features of the vertex and fragment
shaders and how to actually draw graphics with them. With this basic structure in mind, you
then constructed several sample programs that drew simple shapes composed of points on the
screen.
This chapter builds on those basics by exploring how to draw more complex shapes and how to
manipulate those shapes in 3D space. In particular, this chapter looks at

• The critical role of triangles in 3DCG and WebGL’s support for drawing triangles
• Using multiple triangles to draw other basic shapes
• Basic transformations that move, rotate, and scale triangles using simple equations
• How matrix operations make transformations simple
By the end of this chapter, you will have a comprehensive understanding of WebGL’s support for
drawing basic shapes and how to use matrix operations to manipulate those shapes. Chapter 4,
“More Transformations and Basic Animation,” then builds on this knowledge to explore simple
animations.

Drawing Multiple Points
As you are probably aware, 3D models are actually made from a simple building block:
the humble triangle. For example, looking at the frog in Figure 3.1, the figure on the right
side shows the triangles used to make up the shape, and in particular the three vertices
that make up one triangle of the head. So, although this game character has a complex
shape, its basic components are the same as a simple one, except of course for many more
triangles and their associated vertices. By using smaller and smaller triangles, and therefore more and more vertices, you can create more complex or smoother objects. Typically,
a complex shape or game character will consist of tens of thousands of triangles and
their associated vertices. Thus, multiple vertices used to make up triangles are pivotal for
drawing 3D objects.

Figure 3.1

Complex characters are also constructed from multiple triangles

In this section, you explore the process of drawing shapes using multiple vertices.
However, to keep things simple, you’ll continue to use 2D shapes, because the technique
to deal with multiple vertices for a 2D shape is the same as dealing with them for a 3D
object. Essentially, if you can master these techniques for 2D shapes, you can easily understand the examples in the rest of this book that use the same techniques for 3D objects.
As an example of handling multiple vertices, let’s create a program, MultiPoint, that
draws three red points on the screen; remember, three points or vertices make up the
triangle. Figure 3.2 shows a screenshot from Multipoint.

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Drawing and Transforming Triangles

Figure 3.2

MultiPoint

In the previous chapter, you created a sample program, ClickedPoints, that drew multiple points based on mouse clicks. ClickedPoints stored the position of the points in a
JavaScript array (g_points[]) and used the gl.drawArrays() method to draw each point
(Listing 3.1). To draw multiple points, you used a loop that iterated through the array,
drawing each point in turn by passing one vertex at a time to the shader.
Listing 3.1 Drawing Multiple Points as Shown in ClickedPoints.js (Chapter 2)
65
66
67
68
69
70
71

for(var i = 0; i

Clear 

D ra w

Figure 3.3

Processing flowchart for MultiPoints.js

This step is implemented at line 34, the function initVertexBuffers(), in Listing 3.2.
Listing 3.2

MultiPoint.js

1 // MultiPoint.js
2 // Vertex shader program
3 var VSHADER_SOURCE =
4
'attribute vec4 a_Position;\n' +
5
'void main() {\n' +
6
' gl_Position = a_Position;\n' +
7
' gl_PointSize = 10.0;\n' +
8
'}\n';
9
10 // Fragment shader program
...
15
16 function main() {
...

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20
21
27
28
31
32
33
34
35
36
37
38
39
40

// Get the rendering context for WebGL
var gl = getWebGLContext(canvas);
...
// Initialize shaders
if (!initShaders(gl, VSHADER_SOURCE, FSHADER_SOURCE)) {
...
}
// Set the positions of vertices
var n = initVertexBuffers(gl);
if (n < 0) {
console.log('Failed to set the positions of the vertices');
return;
}
// Set the color for clearing 
...
// Clear 

43
...
46
// Draw three points
47
gl.drawArrays(gl.POINTS, 0, n); // n is 3
48 }
49
50 function initVertexBuffers(gl) {
51
var vertices = new Float32Array([
52
0.0, 0.5,
-0.5, -0.5,
0.5, -0.5
53
]);
54
var n = 3; // The number of vertices
55
56
// Create a buffer object
57
var vertexBuffer = gl.createBuffer();
58
if (!vertexBuffer) {
59
console.log('Failed to create the buffer object ');
60
return -1;
61
}
62
63
// Bind the buffer object to target
64
gl.bindBuffer(gl.ARRAY_BUFFER, vertexBuffer);
65
// Write date into the buffer object
66
gl.bufferData(gl.ARRAY_BUFFER, vertices, gl.STATIC_DRAW);
67
68
var a_Position = gl.getAttribLocation(gl.program, 'a_Position');
...
73
// Assign the buffer object to a_Position variable
74
gl.vertexAttribPointer(a_Position, 2, gl.FLOAT, false, 0, 0);

Drawing Multiple Points

71

75
76
// Enable the assignment to a_Position variable
77
gl.enableVertexAttribArray(a_Position);
78
79
return n;
80 }

The new function initVertexBuffers() is defined at line 50 and used at line 34 to set
up the vertex buffer object. The function stores multiple vertices in the buffer object and
then completes the preparations for passing it to a vertex shader:
33 // Set the positions of vertices
34 var n = initVertexBuffers(gl);

The return value of this function is the number of vertices being drawn, stored in the variable n. Note that in case of error, n is negative.
As in the previous examples, the drawing operation is carried out using a single call to
gl.drawArrays() at Line 48. This is similar to ClickedPoints.js except that n is passed as
the third argument of gl.drawArrays() rather than the value 1:
46
47

// Draw three points
gl.drawArrays(gl.POINTS, 0, n); // n is 3

Because you are using a buffer object to pass multiple vertices to a vertex shader in initVertexBuffers(), you need to specify the number of vertices in the object as the third
parameter of gl.drawArrays() so that WebGL then knows to draw a shape using all the
vertices in the buffer object.

Using Buffer Objects
As indicated earlier, a buffer object is a mechanism provided by the WebGL system that
provides a memory area allocated in the system (see Figure 3.4) that holds the vertices you
want to draw. By creating a buffer object and then writing the vertices to the object, you
can pass multiple vertices to a vertex shader through one of its attribute variables.
attribute variable

Ja va S cript
function main() {
var vertices =
new Float32Array([
0.0, 0.5,
-0.5, -0.5,
0.5, -0.5
]);
gl.bufferata(.., vertices,);

Figure 3.4

72

Buffer Object

0.0, 0.5
-0.5, -0.5
0.5, -0.5

Vertex Shader
0.0, 0.5,
-0.5, -0.5,
0.5, -0.5

We b G L
S y s te m

Passing multiple vertices to a vertex shader by using a buffer object

CHAPTER 3

Drawing and Transforming Triangles

In the sample program, the data (vertex coordinates) written into a buffer object is defined
as a special JavaScript array (Float32Array) as follows. We will explain this special array in
detail later, but for now you can think of it as a normal array:
51
52
53

var vertices = new Float32Array([
0.0, 0.5,
-0.5, -0.5,
0.5, -0.5
]);

There are five steps needed to pass multiple data values to a vertex shader through a buffer
object. Because WebGL uses a similar approach when dealing with other objects such as
texture objects (Chapter 4) and framebuffer objects (Chapter 8, “Lighting Objects”), let’s
explore these in detail so you will be able to apply the knowledge later:
1. Create a buffer object (gl.createBuffer()).
2. Bind the buffer object to a target (gl.bindBuffer()).
3. Write data into the buffer object (gl.bufferData()).
4. Assign the buffer object to an attribute variable (gl.vertexAttribPointer()).
5. Enable assignment (gl.enableVertexAttribArray()).
Figure 3.5 illustrates the five steps.
(4)

(5)
attribute variable

Ja va S cript

(1)(2)
function main() {
var vertices =
new Float32Array([
0.0, 0.5,
-0.5, -0.5,
(3)
0.5, -0.5
]);
gl.bufferata(.., vertices,);

Figure 3.5

Buffer Object

0.0, 0.5
-0.5, -0.5
0.5, -0.5

Vertex Shader
0.0, 0.5,
-0.5, -0.5,
0.5, -0.5

We b G L
S y s te m

The five steps to pass multiple data values to a vertex shader using a buffer object

The code performing the steps in the sample program in Listing 3.2 is as follows:
56
57
58
59
60
61
62
63

// Create a buffer object
var vertexBuffer = gl.createBuffer();
if (!vertexBuffer) {
console.log('Failed to create a buffer object');
return -1;
}

<- (1)

// Bind the buffer object to a target

<- (2)

Drawing Multiple Points

73

64
65
66
67
68
73
74
75
76
77

gl.bindBuffer(gl.ARRAY_BUFFER, vertexBuffer);
// Write date into the buffer object
gl.bufferData(gl.ARRAY_BUFFER, vertices, gl.STATIC_DRAW);

<- (3)

var a_Position = gl.getAttribLocation(gl.program, 'a_Position');
...
// Assign the buffer object to a_Position variable
<- (4)
gl.vertexAttribPointer(a_Position, 2, gl.FLOAT, false, 0, 0);
// Enable the assignment to a_Position variable
gl.enableVertexAttribArray(a_Position);

<- (5)

Let’s start with the first three steps (1–3), from creating a buffer object to writing data
(vertex coordinates in this example) to the buffer, explaining the methods used within
each step.

Create a Buffer Object (gl.createBuffer())
Before you can use a buffer object, you obviously need to create the buffer object. This is
the first step, and it’s carried out at line 57:
57

var vertexBuffer = gl.createBuffer();

You use the gl.createBuffer() method to create a buffer object within the WebGL
system. Figure 3.6 shows the internal state of the WebGL system. The upper part of the
figure shows the state before executing the method, and the lower part is after execution.
As you can see, when the method is executed, it results in a single buffer object being
created in the WebGL system. The keywords gl.ARRAY_BUFFER and gl.ELEMENT_ARRAY_
BUFFER in the figure will be explained in the next section, so you can ignore them for
now.
attribute variable

JavaS cript

g l .A R R A Y _ B U F F E R

g l . EL EM ENT_A RRA Y_B UFFER

Vertex Shader

fu n c tio n m a in () {
v ar gl= g etWebGL…
…

attribute variable

JavaS cript

g l .A R R A Y _ B U F F E R

fu n c tio n m a in () {
v ar gl= g etWebGL…
…
g l.c r e a te B u ffe r ( … ) ;

Figure 3.6

74

g l . EL EM ENT_A RRA Y_B UFFER

Vertex Shader
B u ffe r O b je c t

Create a buffer object

CHAPTER 3

Drawing and Transforming Triangles

The following shows the specification of gl.createBuffer().

gl.createBuffer()
Create a buffer object.
Return value

Errors

non-null

The newly created buffer object.

null

Failed to create a buffer object.

None

The corresponding method gl.deleteBuffer() deletes the buffer object created by
gl.createBuffer().

gl.deleteBuffer(buffer)
Delete the buffer object specified by buffer.
Parameters

buffer

Return Value

None

Errors

None

Specifies the buffer object to be deleted.

Bind a Buffer Object to a Target (gl.bindBuffer())
After creating a buffer object, the second step is to bind it to a “target.” The target tells
WebGL what type of data the buffer object contains, allowing it to deal with the contents
correctly. This binding process is carried out at line 64 as follows:
64 gl.bindBuffer(gl.ARRAY_BUFFER, vertexBuffer);

The specification of gl.bindBuffer() is as follows.

gl.bindBuffer(target, buffer)
Enable the buffer object specified by buffer and bind it to the target.
Parameters

Target can be one of the following:
gl.ARRAY_BUFFER

Specifies that the buffer object contains vertex data.

Drawing Multiple Points

75

gl.ELEMENT_
ARRAY_BUFFER

buffer

Specifies that the buffer object contains index values
pointing to vertex data. (See Chapter 6, “The OpenGL
ES Shading Language [GLSL ES].)
Specifies the buffer object created by a previous call to
gl.createBuffer().
When null is specified, binding to the target is
disabled.

Return Value

None

Errors

INVALID_ENUM

target is none of the above values. In this case, the
current binding is maintained.

In the sample program in this section, gl.ARRAY_BUFFER is specified as the target to store
vertex data (positions) in the buffer object. After executing line 64, the internal state in
the WebGL system changes, as shown in Figure 3.7.
JavaScript
fu n c tio n m a in () {
v ar gl= g etWebGL…
…
g l. b i n d B u f f e r ( … ) ;

Figure 3.7

attribute variable
g l .A R R A Y _ B U F F E R

g l . EL EM ENT_A RRA Y_B UFFER

Vertex Shader
B u ffe r O b je c t

Bind a buffer object to a target

The next step is to write data into the buffer object. Note that because you won’t be using
the gl.ELEMENT_ARRAY_BUFFER until Chapter 6, it’ll be removed from the following figures
for clarity.

Write Data into a Buffer Object (gl.bufferData())
Step 3 allocates storage and writes data to the buffer. You use gl.bufferData() to do this,
as shown at line 66:
66

gl.bufferData(gl.ARRAY_BUFFER, vertices, gl.STATIC_DRAW);

This method writes the data specified by the second parameter (vertices) into the buffer
object bound to the first parameter (gl.ARRAY_BUFFER). After executing line 66, the internal state of the WebGL system changes, as shown in Figure 3.8.

76

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Drawing and Transforming Triangles

attribute variable

JavaS cript
function main() {
var vertices =
new Float32Array([
0.0, 0.5,
-0.5, -0.5,
0.5, -0.5
]);
gl.bufferData(.., vertices,);

Figure 3.8

Buffer Object

Vertex Shader

0.0, 0.5
-0.5, -0.5
0.5, -0.5

We b G L
S y s te m

Allocate storage and write data into a buffer object

You can see in this figure that the vertex data defined in your JavaScript program is
written to the buffer object bound to gl.ARRAY_BUFFER. The following table shows the
specification of gl.bufferData().

gl.bufferData(target, data, usage)
Allocate storage and write the data specified by data to the buffer object bound to target.
Parameters

target

Specifies gl.ARRAY_BUFFER or gl.ELEMENT_ARRAY_BUFFER.

data

Specifies the data to be written to the buffer object (typed
array; see the next section).

usage

Specifies a hint about how the program is going to use
the data stored in the buffer object. This hint helps WebGL
optimize performance but will not stop your program from
working if you get it wrong.

Return value

None

Errors

INVALID_ENUM

gl.STATIC_
DRAW

The buffer object data will be specified once
and used many times to draw shapes.

gl.STREAM_
DRAW

The buffer object data will be specified once
and used a few times to draw shapes.

gl.DYNAMIC_
DRAW

The buffer object data will be specified repeatedly and used many times to draw shapes.

target is none of the preceding constants

Now, let us examine what data is passed to the buffer object using gl.bufferData().
This method uses the special array vertices mentioned earlier to pass data to the vertex
shader. The array is created at line 51 using the new operator with the data arranged as
, , and so on:

Drawing Multiple Points

77

51
52
53
54

var vertices = new Float32Array([
0.0, 0.5,
-0.5, -0.5,
0.5, -0.5
]);
var n = 3; // The number of vertices

As you can see in the preceding code snippet, you are using the Float32Array object
instead of the more usual JavaScript Array object to store the data. This is because the
standard array in JavaScript is a general-purpose data structure able to hold both numeric
data and strings but isn’t optimized for large quantities of data of the same type, such as
vertices. To address this issue, the typed array, of which one example is Float32Array,
has been introduced.

Typed Arrays
WebGL often deals with large quantities of data of the same type, such as vertex coordinates and colors, for drawing 3D objects. For optimization purposes, a special type of array
(typed array) has been introduced for each data type. Because the type of data in the
array is known in advance, it can be handled efficiently.
Float32Array at line 51 is an example of a typed array and is generally used to store
vertex coordinates or colors. It’s important to remember that a typed array is expected
by WebGL and is needed for many operations, such as the second parameter data of
gl.bufferData().

Table 3.1 shows the different typed arrays available. The third column shows the corresponding data type in C as a reference for those of you familiar with the C language.
Table 3.1

Typed Arrays Used in WebGL

Typed Array

Number of Bytes per Element

Description (C Types)

Int8Array

1

8-bit signed integer (signed char)

Uint8Array

1

8-bit unsigned integer (unsigned char)

Int16Array

2

16-bit signed integer (signed short)

Uint16Array

2

16-bit unsigned integer (unsigned short)

Int32Array

4

32-bit signed integer (signed int)

Uint32Array

4

32-bit unsigned integer (unsigned int)

Float32Array

4

32-bit floating point number (float)

Float64Array

8

64-bit floating point number (double)

Like JavaScript, these typed arrays have a set of methods, a property, and a constant available that are shown in Table 3.2. Note that, unlike the standard Array object in JavaScript,
the methods push() and pop() are not supported.

78

CHAPTER 3

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Table 3.2

Methods, Property, Constant of Typed Arrays

Methods, Properties, and Constants

Description

get(index)

Get the index-th element

set(index, value)

Set value to the index-th element

set(array, offset)

Set the elements of array from offset-th element

length

The length of the array

BYTES_PER_ELEMENT

The number of bytes per element in the array

Just like standard arrays, the new operator creates a typed array and is passed the array
data. For example, to create Float32Array vertices, you could pass the array [0.0, 0.5,
-0.5, -0.5, 0.5, -0.5], which represents a set of vertices. Note that the only way to
create a typed array is by using the new operator. Unlike the Array object, the [] operator
is not supported:
51
52
53

var vertices = new Float32Array([
0.0, 0.5,
-0.5, -0.5,
0.5, -0.5
]);

In addition, just like a normal JavaScript array, an empty typed array can be created by
specifying the number of elements of the array as an argument. For example:
var vertices = new Float32Array(4);

With that, you’ve completed the first three steps of the process to set up and use a buffer
(that is, creating a buffer object in the WebGL system, binding the buffer object to a
target, and then writing data into the buffer object). Let’s now look at how to actually use
the buffer, which takes place in steps 4 and 5 of the process.

Assign the Buffer Object to an Attribute Variable
(gl.vertexAttribPointer())
As explained in Chapter 2, you can use gl.vertexAttrib[1234]f() to assign data to an
attribute variable. However, these methods can only be used to assign a single data value
to an attribute variable. What you need here is a way to assign an array of values—the
vertices in this case—to an attribute variable.
gl.vertexAttribPointer() solves this problem and can be used to assign a buffer object
(actually a reference or handle to the buffer object) to an attribute variable. This can be
seen at line 74 when you assign a buffer object to the attribute variable a_Position:
74

gl.vertexAttribPointer(a_Position, 2, gl.FLOAT, false, 0, 0);

The specification of gl.vertexAttribPointer() is as follows.

Drawing Multiple Points

79

gl.vertexAttribPointer(location, size, type, normalized, stride,
offset)
Assign the buffer object bound to gl.ARRAY_BUFFER to the attribute variable specified by
location.
Parameters

location

Specifies the storage location of an attribute variable.

size

Specifies the number of components per vertex in the buffer object
(valid values are 1 to 4). If size is less than the number of components required by the attribute variable, the missing components
are automatically supplied just like gl.vertexAttrib[1234]f().
For example, if size is 1, the second and third components will be
set to 0, and the fourth component will be set to 1.

type

Specifies the data format using one of the following:

gl.UNSIGNED_BYTE

unsigned byte

for Uint8Array

gl.SHORT

signed short integer

for Int16Array

gl.UNSIGNED_SHORT

unsigned short integer

for Uint16Array

gl.INT

signed integer

for Int32Array

gl.UNSIGNED_INT

unsigned integer

for Uint32Array

gl.FLOAT

floating point number

for Float32Array

normalized

Either true or false to indicate whether nonfloating data should
be normalized to [0, 1] or [–1, 1].

stride

Specifies the number of bytes between different vertex data
elements, or zero for default stride (see Chapter 4).

offset

Specifies the offset (in bytes) in a buffer object to indicate what
number-th byte the vertex data is stored from. If the data is stored
from the beginning, offset is 0.

Return value None
Errors

80

INVALID_OPERATION

There is no current program object.

INVALID_VALUE

location is greater than or equal to the maximum
number of attribute variables (8, by default). stride or
offset is a negative value.

CHAPTER 3

Drawing and Transforming Triangles

So, after executing this fourth step, the preparations are nearly completed in the WebGL
system for using the buffer object at the attribute variable specified by location. As you can
see in Figure 3.9, although the buffer object has been assigned to the attribute variable,
WebGL requires a final step to “enable” the assignment and make the final connection.
attribute variable

JavaScript

g l .A R R A Y _ B U F F E R

fu n c tio n m a in () {
v ar gl= g etWebGL…
…
g l.v e r te x A ttr ib P o in te r ( … ) ;
…

Figure 3.9

Vertex Shader
B u ffe r O b je c t

0.0, 0.5
-0.5, -0.5

Assign a buffer object to an attribute variable

The fifth and final step is to enable the assignment of the buffer object to the attribute
variable.

Enable the Assignment to an Attribute Variable
(gl.enableVertexAttribArray())
To make it possible to access a buffer object in a vertex shader, we need to enable the
assignment of the buffer object to an attribute variable by using gl.enableVertexAttribArray() as shown in line 77:
77

gl.enableVertexAttribArray(a_Position);

The following shows the specification of gl.enableVertexAttribArray(). Note that we are
using the method to handle a buffer even though the method name suggests it’s only for
use with “vertex arrays.” This is not a problem and is simply a legacy from OpenGL.

gl.enableVertexAttribArray(location)
Enable the assignment of a buffer object to the attribute variable specified by location.
Parameters

location

Return value

None

Errors

INVALID_VALUE

Specifies the storage location of an attribute variable.

location is greater than or equal to the maximum number
of attribute variables (8 by default).

When you execute gl.enableVertexAttribArray() specifying an attribute variable that
has been assigned a buffer object, the assignment is enabled, and the unconnected line is
connected as shown in Figure 3.10.

Drawing Multiple Points

81

attribute variable

JavaScript

g l .A R R A Y _ B U F F E R

Vertex Shader

fu n c tio n m a in () {
v ar gl= g etWebGL…
…
gl.vertexAttribArray(...);
…

Figure 3.10

B u ffe r O b je c t

0.0, 0.5
-0.5, -0.5

Enable the assignment of a buffer object to an attribute variable

You can also break this assignment (disable it) using the method
gl.disableVertexAttribArray().

gl.disableVertexAttribArray(location)
Disable the assignment of a buffer object to the attribute variable specified by location.
Parameters

location

Return Value

None

Errors

INVALID_VALUE

Specifies the storage location of an attribute variable.

location is greater than or equal to the maximum number
of attribute variables (8 by default).

Now, everything is set! All you need to do is run the vertex shader, which draws the
points using the vertex coordinates specified in the buffer object. As in Chapter 2, you
will use the method gl.drawArrays, but because you are drawing multiple points, you will
actually use the second and third parameters of gl.drawArrays().
Note that after enabling the assignment, you can no longer use gl.vertexAttrib[1234]
f() to assign data to the attribute variable. You have to explicitly disable the assignment
of a buffer object. You can’t use both methods simultaneously.

The Second and Third Parameters of gl.drawArrays()
Before entering into a detailed explanation of these parameters, let’s take a look at the
specification of gl.drawArrays() that was introduced in Chapter 2. Following is a recap of
the method with only the relevant parts of the specification shown.

82

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Drawing and Transforming Triangles

gl.drawArrays(mode, first, count)
Execute a vertex shader to draw shapes specified by the mode parameter.
Parameters mode

Specifies the type of shape to be drawn. The following symbolic
constants are accepted: gl.POINTS, gl.LINES, gl.LINE_STRIP, gl.
LINE_LOOP, gl.TRIANGLES, gl.TRIANGLE_STRIP, and gl.TRIANGLE_
FAN.

first

Specifies what number-th vertex is used to draw from (integer).

count

Specifies the number of vertices to be used (integer).

In the sample program this method is used as follows:
47

gl.drawArrays(gl.POINTS, 0, n); // n is 3

As in the previous examples, because you are simply drawing three points, the first parameter is still gl.POINTS. The second parameter first is set to 0 because you want to draw
from the first coordinate in the buffer. The third parameter count is set to 3 because you
want to draw three points (in line 47, n is 3).
When your program runs line 47, it actually causes the vertex shader to be executed count
(three) times, sequentially passing the vertex coordinates stored in the buffer object via
the attribute variable into the shader (Figure 3.11).
Note that for each execution of the vertex shader, 0.0 and 1.0 are automatically supplied
to the z and w components of a_Position because a_Position requires four components
(vec4) and you are supplying only two.
Remember that at line 74, the second parameter size of gl.vertexAttribPointer() is set
to 2. As just discussed, the second parameter indicates how many coordinates per vertex
are specified in the buffer object and, because you are only specifying the x and y coordinates in the buffer, you set the size value to 2:
74

gl.vertexAttribPointer(a_Position, 2, gl.FLOAT, false, 0, 0);

After drawing all points, the content of the color buffer is automatically displayed in the
browser (bottom of Figure 3.11), resulting in our three red points, as shown in Figure 3.2.

Drawing Multiple Points

83

First Execution
JavaScript

gl.ARRAY_BUFFER

function main() {
…
var gl=getWebGL…
…
gl.drawArrays( …, 0, n);

attribute a_Position;

Vertex Shader
Buffer Object

0.0, 0.5
-0.5, -0.5
0.5, -0.5

Fragment
Shader

Color Buffer

(0.0, 0.5, 0.0, 1.0)
gl_Position = a_Position;

gl_FragColor =
vec4(...);

Second Execution
JavaScript

gl.ARRAY_BUFFER

function main() {
…
var gl=getWebGL…
…
gl.drawArrays( …, 0, n);

attribute a_Position;

Vertex Shader
Buffer Object

0.0, 0.5
-0.5, -0.5
0.5, -0.5

Fragment
Shader

Color Buffer

(-0.5, -0.5, 0.0, 1.0)
gl_Position = a_Position;

gl_FragColor =
vec4(...);

Third Execution
JavaScript

gl.ARRAY_BUFFER

attribute a_Position;

Browser
function main() {
…
var gl=getWebGL…
…
gl.drawArrays(…, 0, n);

Figure 3.11

Vertex Shader
Buffer Object

0.0, 0.5
-0.5, -0.5
0.5, -0.5

Fragment
Shader

Color Buffer

(0.5, -0.5, 0.0, 1.0)
gl_Position = a_Position;

gl_FragColor =
vec4(...);

How the data in a buffer object is passed to a vertex shader during execution

Experimenting with the Sample Program
Let’s experiment with the sample program to better understand how gl.drawArrays()
works by modifying the second and third parameters. First, let’s specify 1 as the third
argument for count at line 47 instead of our variable n (set to 3) as follows:
47

gl.drawArrays(gl.POINTS, 0, 1);

In this case, the vertex shader is executed only once, and a single point is drawn using the
first vertex in the buffer object.
If you now specify 1 as the second argument, only the second vertex is used to draw
a point. This is because you are telling WebGL that you want to start drawing from
the second vertex and you only want to draw one vertex. So again, you will see only a
single point, although this time it is the second vertex coordinates that are shown in the
browser:
47

84

gl.drawArrays(gl.POINTS, 1, 1);

CHAPTER 3

Drawing and Transforming Triangles

This gives you a quick feel for the role of the parameters first and count. However, what
will be happen if you change the first parameter mode? The next section explores the first
parameter in more detail.

Hello Triangle
Now that you’ve learned the basic techniques to pass multiple vertex coordinates to a
vertex shader, let’s try to draw other shapes using multiple vertex coordinates. This section
uses a sample program HelloTriangle, which draws a single 2D triangle. Figure 3.12
shows a screenshot of HelloTriangle.

Figure 3.12

HelloTriangle

Sample Program (HelloTriangle.js)
Listing 3.3 shows HelloTriangle.js, which is almost identical to MultiPoint.js used in
the previous section with two critical differences.
Listing 3.3 HelloTriangle.js
1 // HelloTriangle.js
2 // Vertex shader program
3 var VSHADER_SOURCE =
4
'attribute vec4 a_Position;\n' +
5
'void main() {\n' +
6
' gl_Position = a_Position;\n' +
7
'}\n';
8

Hello Triangle

85

9 // Fragment shader program
10 var FSHADER_SOURCE =
11
'void main() {\n' +
12
' gl_FragColor = vec4(1.0, 0.0, 0.0, 1.0);\n' +
13
'}\n';
14
15 function main() {
...
19
// Get the rendering context for WebGL
20
var gl = getWebGLContext(canvas);
...
26
// Initialize shaders
27
if (!initShaders(gl, VSHADER_SOURCE, FSHADER_SOURCE)) {
...
30
}
31
32
// Set the positions of vertices
33
var n = initVertexBuffers(gl);
...
39
// Set the color for clearing 
...
45
// Draw a triangle
46
gl.drawArrays(gl.TRIANGLES, 0, n);
47 }
48
49 function initVertexBuffers(gl) {
50
var vertices = new Float32Array([
51
0.0, 0.5,
-0.5, -0.5,
0.5, -0.5
52
]);
53
var n = 3; // The number of vertices
...
78
return n;
79 }

The two differences from MultiPoint.js are

• The line to specify the size of a point gl_PointSize

= 10.0; has been removed from
the vertex shader. This line only has an effect when you are drawing a point.

• The first parameter of gl.drawArrays() has been changed from gl.POINTS to
gl.TRIANGLES at line 46.

The first parameter, mode, of gl.drawArrays() is powerful and provides the ability to draw
various shapes. Let’s take a look.

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Basic Shapes
By changing the argument we use for the first parameter, mode, of gl.drawArrays(), we
can change the meaning of line 46 into “execute the vertex shader three times (n is 3),
and draw a triangle using the three vertices in the buffer, starting from the first vertex
coordinate”:
46

gl.drawArrays(gl.TRIANGLES, 0, n);

In this case, the three vertices in the buffer object are no longer individual points, but
become three vertices of a triangle.
The WebGL method gl.drawArrays() is both powerful and flexible, allowing you to
specify seven different types of basic shapes as the first argument. These are explained in
more detail in Table 3.3. Note that v0, v1, v2 ... indicates the vertices specified in a buffer
object. The order of vertices affects the drawing of the shape.
The shapes in the table are the only ones that WebGL can draw directly, but they are the
basics needed to construct complex 3D graphics. (Remember the frog at the start of this
chapter.)
Table 3.3

Basic Shapes Available in WebGL

Basic
Shape

Mode

Description

Points

gl.POINTS

A series of points. The points are drawn at v0, v1, v2 ...

Line
segments

gl.LINES

A series of unconnected line segments. The individual lines
are drawn between vertices given by (v0, v1), (v2, v3), (v4,
v5)... If the number of vertices is odd, the last one is ignored.

Line strips gl.LINE_STRIP

A series of connected line segments. The line segments are
drawn between vertices given by (v0, v1), (v1, v2), (v2, v3), ...
The first vertex becomes the start point of the first line, the
second vertex becomes the end point of the first line and the
start point of the second line, and so on. The i-th (i> 1) vertex
becomes the start point of the i-th line and the end point of
the i-1-th line. (The last vertex becomes the end point of the
last line.)

Line loops

gl.LINE_LOOP

A series of connected line segments. In addition to the lines
drawn by gl.LINE_STRIP, the line between the last vertex and
the first vertex is drawn. The line segments drawn are (v0, v1),
(v1, v2), ..., and (vn, v0). vn is the last vertex.

Triangles

gl.TRIANGLES

A series of separate triangles. The triangles given by vertices
(v0, v1, v2), (v3, v4, v5), ... are drawn. If the number of vertices is not a multiple of 3, the remaining vertices are ignored.

Hello Triangle

87

Basic
Shape

Mode

Description

Triangle
strips

gl.TRIANGLE_
STRIP

A series of connected triangles in strip fashion. The first three
vertices form the first triangle and the second triangle is
formed from the next vertex and one of the sides of the first
triangle. The triangles are drawn given by (v0, v1, v2), (v2, v1,
v3), (v2, v3, v4) ... (Pay attention to the order of vertices.)

Triangle
fans

gl.TRIANGLE_
FAN

A series of connected triangles sharing the first vertex in fanlike fashion. The first three vertices form the first triangle and
the second triangle is formed from the next vertex, one of the
sides of the first triangle, and the first vertex. The triangles are
drawn given by (v0, v1, v2), (v0, v2, v3), (v0, v3, v4), ...

Figure 3.13 shows these basic shapes.
v0

v4

v2

v1

v3

v5

g l.P O IN T S
v0

v4

v2

v1

v0

v3
g l.L IN E S
v2

v0

v0

v1
v3
v5
g l.L IN E _ S T R IP

v5

v0

v5

v4

v2

v2

v1

v5
v3
g l.L IN E _ L O O P
v3

v4

v2

v4

v1

v3

v4

v1

v3

v5
v5

g l.T R IA N G L E S

v4

v2

v1
g l.T R IA N G L E _ F A N

g l.T R IA N G L E _ S T R IP
v0

Figure 3.13

Basic shapes available in WebGL

As you can see from the figure, WebGL can draw only three types of shapes: a point, a
line, and a triangle. However, as explained at the beginning of this chapter, spheres to
cubes to 3D monsters to humanoid characters in a game can be constructed from small
triangles. Therefore, you can use these basic shapes to draw anything.

88

CHAPTER 3

Drawing and Transforming Triangles

Experimenting with the Sample Program
To examine what will happen when using gl.LINES, gl.LINE_STRIP, and gl.LINE_LOOP,
let’s change the first argument of gl.drawArrays() as shown next. The name of each
sample program is HelloTriangle_LINES, HelloTriangle_LINE_STRIP, and HelloTriangle_
LINE_LOOP, respectively:
46
46
46

gl.drawArrays(gl.LINES, 0, n);
gl.drawArrays(gl.LINE_STRIP, 0, n);
gl.drawArrays(gl.LINE_LOOP, 0, n);

Figure 3.14 shows a screenshot of each program.

Figure 3.14

gl.LINES, gl.LINE_STRIP, and gl.LINE_LOOP

As you can see, gl.LINES draws a line using the first two vertices and does not use the
last vertex, whereas gl.LINE_STRIP draws two lines using the first three vertices. Finally,
gl.LINE_LOOP draws the lines in the same manner as gl.LINE_STRIP but then “loops”
between the last vertex and the first vertex and makes a triangle.

Hello Rectangle (HelloQuad)
Let’s use this basic way of drawing triangles to draw a rectangle. The name of the sample
program is HelloQuad, and Figure 3.15 shows a screenshot when it’s loaded into your
browser.
Figure 3.16 shows the vertices of the rectangle. Of course, the number of vertices is four
because it is a rectangle. As explained in the previous section, WebGL cannot draw a rectangle directly, so you need to divide the rectangle into two triangles (v0, v1, v2) and (v2,
v1, v3) and then draw each one using gl.TRIANGLES, gl.TRIANGLE_STRIP, or gl.TRIANGLE_
FAN. In this example, you’ll use gl.TRIANGLE_STRIP because it only requires you to specify
four vertices. If you were to use gl.TRIANGLES, you would need to specify a total of six.

Hello Triangle

89

Figure 3.15

HelloQuad

v0
(-0.5, 0.5, 0.0)

y

v2
(0.5, 0.5, 0.0)

x

v3
(0.5, -0.5, 0.0)

v1
(-0.5, -0.5, 0.0)

Figure 3.16

The four vertex coordinates of the rectangle

Basing the example on HelloTriangle.js, you need to add an extra vertex coordinate
at line 50. Pay attention to the order of vertices; otherwise, the draw command will not
execute correctly:
50
51
52

var vertices = new Float32Array([
-0.5, 0.5,
-0.5, -0.5,
0.5, 0.5,
]);

0.5, -0.5

Because you’ve added a fourth vertex, you need to change the number of vertices from 3
to 4 at line 53:
53

90

var n = 4; // The number of vertices

CHAPTER 3

Drawing and Transforming Triangles

Then, by modifying line 46 as follows, your program will draw a rectangle in the browser:
46

gl.drawArrays(gl.TRIANGLE_STRIP, 0, n);

Experimenting with the Sample Program
Now that you have a feel for how to use gl.TRIANGLE_STRIP, let’s change the first
parameter of gl.drawArrays() to gl.TRIANGLE_FAN. The name of the sample program is
HelloQuad_FAN:
46

gl.drawArrays(gl.TRIANGLE_FAN, 0, n);

Figure 3.17 show a screenshot of HelloQuad_FAN. In this case, we can see the ribbon-like
shape on the screen.
v0

v2

v1

v3

gl.TRIANGLE_STRIP
v0

v2

v1

v3

gl.TRIANGLE_FAN

Figure 3.17

HelloQuad_FAN

Looking at the order of vertices and the triangles drawn by gl.TRIANGLE_FAN shown on
the right side of Figure 3.17, you can see why the result became a ribbon-like shape.
Essentially, gl.TRIANGLE_FAN causes WebGL to draw a second triangle that shares the first
vertex (v0), and this second triangle overlaps the first, creating the ribbon-like effect.

Moving, Rotating, and Scaling
Now that you understand the basics of drawing shapes like triangles and rectangles, let’s
take another step and try to move (translate), rotate, and scale the triangle and display the
results on the screen. These operations are called transformations (affine transformations). This section introduces some math to explain each transformation and help you
to understand how each operation can be realized. However, when you write your own
programs, you don’t need the math; instead, you can use one of several convenient libraries, explained in the next section, that handle the math for you.

Moving, Rotating, and Scaling

91

If you find reading this section and in particular the math too much on first read, it’s
okay to skip it and return later. Or, if you already know that transformations can be
written using a matrix, you can skip this section as well.
First, let’s write a sample program, TranslatedTriangle, that moves a triangle 0.5 units
to the right and 0.5 units up. You can use the triangle you drew in the previous section.
The right direction means the positive direction of the x-axis, and the up direction means
the positive direction of the y-axis. (See the coordinate system in Chapter 2.) Figure 3.18
shows TranslatedTriangle.

Figure 3.18

TranslatedTriangle

Translation
Let us examine what kind of operations you need to apply to each vertex coordinate
of a shape to translate (move) the shape. Essentially, you just need to add a translation
distance for each direction (x and y) to each component of the coordinates. Looking at
Figure 3.19, the goal is to translate the point p (x, y, z) to the point p' (x', y', z'), so the
translation distance for the x, y, and z direction is Tx, Ty, and Tz, respectively. In this
figure, Tz is 0.
To determine the coordinates of p', you simply add the T values, as shown in
Equation 3.1.
Equation 3.1
x' = x + Tx
y' = y + Ty
z' = z + Tz

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Drawing and Transforming Triangles

y

p ' ( x ' , y ' , z' )

Tx

Ty

x

p(x , y , z)
Figure 3.19

Calculating translation distances

These simple equations can be implemented in a WebGL program just by adding each
constant value to each vertex coordinate. You’ve probably realized already that because
they are a per-vertex operation, you need to implement the operations in a vertex
shader. Conversely, they clearly aren’t a per-fragment operation, so you don’t need to
worry about the fragment shader.
Once you understand this explanation, implementation is easy. You need to pass the
translation distances Tx, Ty, and Tz to the vertex shader, apply Equation 3.1 using the
distances, and then assign the result to gl_Position. Let’s look at a sample program that
does this.

Sample Program (TranslatedTriangle.js)
Listing 3.4 shows TranslatedTriangle.js, in which the vertex shader is partially modified to carry out the translation operation. However, the fragment shader is the same as
in HelloTriangle.js in the previous section. To support the modification to the vertex
shader, some extra code is added to the main() function in the JavaScript.
Listing 3.4 TranslatedTriangle.js
1
2
3
4
5
6
7
8
9
10

// TranslatedTriangle.js
// Vertex shader program
var VSHADER_SOURCE =
'attribute vec4 a_Position;\n' +
'uniform vec4 u_Translation;\n' +
'void main() {\n' +
' gl_Position = a_Position + u_Translation;\n' +
'}\n';
// Fragment shader program
...

Moving, Rotating, and Scaling

93

16 // The translation distance for x, y, and z direction
17 var Tx = 0.5, Ty = 0.5, Tz = 0.0;
18
19 function main() {
...
23
// Get the rendering context for WebGL
24
var gl = getWebGLContext(canvas);
...
30
// Initialize shaders
31
if (!initShaders(gl, VSHADER_SOURCE, FSHADER_SOURCE)) {
...
34 }
35
36
// Set the positions of vertices
37
var n = initVertexBuffers(gl);
...
43
// Pass the translation distance to the vertex shader
44
var u_Translation = gl.getUniformLocation(gl.program, 'u_Translation');
...
49
gl.uniform4f(u_Translation, Tx, Ty, Tz, 0.0);
50
51
// Set the color for clearing 
...
57
// Draw a triangle
58
gl.drawArrays(gl.TRIANGLES, 0, n);
59 }
60
61 function initVertexBuffers(gl) {
62
var vertices = new Float32Array([
63
0.0.0, 0.5,
-0.5, -0.5,
0.5, -0.5
64
]);
65
var n = 3; // The number of vertices
...
90
return n;
93 }

First, let’s examine main() in JavaScript. Line 17 defines the variables for each translation
distance of Equation 3.1:
17 var Tx = 0.5, Ty = 0.5, Tz = 0.0;

Because Tx, Ty, and Tz are fixed (uniform) values for all vertices, you use the uniform variable u_Translation to pass them to a vertex shader. Line 44 retrieves the storage location
of the uniform variable, and line 49 assigns the data to the variable:

94

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Drawing and Transforming Triangles

44
49

var u_Translation = gl.getUniformLocation(gl.program, 'u_Translation');
...
gl.uniform4f(u_Translation, Tx, Ty, Tz, 0.0);

Note that gl.uniform4f() requires a homogenous coordinate, so we supply a fourth argument (w) of 0.0. This will be explained in more detail later in this section.
Now, let’s take a look at the vertex shader that uses this translation data. As you can see,
the uniform variable u_Translation in the shader, to which the translation distances
are passed, is defined as type vec4 at line 5. This is because you want to add the components of u_Translation to the vertex coordinates passed to a_Position (as defined by
Equation 3.1) and then assign the result to the variable gl_Position, which has type
vec4. Remember, per Chapter 2, that the assignment operation in GLSL ES is only allowed
between variables of the same types:
4
5
6
7
8

'attribute vec4 a_Position;\n' +
'uniform vec4 u_Translation;\n' +
'void main() {\n' +
' gl_Position = a_Position + u_Translation;\n' +
'}\n';

After these preparations have been completed, the rest of tasks are straightforward. To
calculate Equation 3.1 within the vertex shader, you just add each translation distance
(Tx, Ty, Tz) passed in u_Translation to each vertex coordinate (x, y, z) passed in
a_Position.
Because both variables are of type vec4, you can use the + operator, which will actually
add the four components simultaneously (see Figure 3.20). This easy addition of vectors is
a feature of GLSL ES and will be explained in more detail in Chapter 6.
v e c 4 a _ P o s itio n

x1

y1

z1

w1

v e c 4 u _ T r a n s la tio n

x2

y2

z2

w2

x1+x2

y1+y2

z1+z2

w1+w2

Figure 3.20

Addition of vec4 variables

Now, we’ll return to the fourth element, (w), of the vector. As explained in Chapter 2, you
need to specify the homogeneous coordinate to gl_Position, which is a four-dimensional
coordinate. If the last component of the homogeneous coordinate is 1.0, the coordinate
indicates the same position as the three-dimensional coordinate. In this case, because the
last component is w1+w2 to ensure that w1+w2 is 1.0, you need to specify 0.0 to the value of
w (the fourth parameter of gl.uniform4f()).

Moving, Rotating, and Scaling

95

Finally, at line 58, gl.drawArrays(gl.TRIANGLES, 0, n) executes the vertex shader. For
each execution, the following three steps are performed:
1. Each vertex coordinate set is passed to a_Position.
2. u_Translation is added to a_Position.
3. The result is assigned to gl_Position.
Once executed, you’ve achieved your goal because each vertex coordinate set is modified (translated), and then the translated shape (in this case, a triangle) is displayed on
the screen. If you now load TranslatedTriangle.html into your browser, you will see the
translated triangle.
Now that you’ve mastered translation (moving), the next step is to look at rotation. The
basic approach to realize rotation is the same as translation, requiring you to manipulate
the vertex coordinates in the vertex shader.

Rotation
Rotation is a little more complex than translation because you have to specify multiple
items of information. The following three items are required:

• Rotation axis (the axis the shape will be rotated around)
• Rotation direction (the direction: clockwise or counterclockwise)
• Rotation angle (the number of degrees the shape will be rotated through)
In this section, to simplify the explanation, you can assume that the rotation is performed
around the z-axis, in a counterclockwise direction, and for β degrees. You can use the
same approach to implement other rotations around the x-axis or y-axis.
In the rotation, if β is positive, the rotation is performed in a counterclockwise direction
around the rotation axis looking at the shape toward the negative direction of the z-axis
(see Figure 3.21); this is called positive rotation. Just as for the coordinate system, your
hand can define the direction of rotation. If you take your right hand and have your
thumb follow the direction of the rotation axis, your fingers show the direction of rotation. This is called the right-hand-rule rotation. As we discussed in Chapter 2, it’s the
default we are using for WebGL in this book.
Now let’s find the expression to calculate the rotation in the same way that you did for
translation. As shown in Figure 3.22, we assume that the point p' (x', y', z') is the β degree
rotated point of p (x, y, z) around the z-axis. Because the rotation is around the z-axis, the
z coordinate does not change, and you can ignore it for now. The explanation is a little
mathematical, so let’s take it a step at a time.

96

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y

Screen

x

z

Figure 3.21

Positive rotation around the z-axis

y

p ' ( x ' , y ' , z' )
β

ｒ

p(x , y , z)

α
x

Figure 3.22

Calculating rotation around the z-axis

In Figure 3.22, r is the distance from the origin to the point p, and α is the rotation angle
from the x-axis to the point. You can use these items of information to represent the coordinates of p, as shown in Equation 3.2.
Equation 3.2
x = r cos α
y = r sin α
Similarly, you can find the coordinate of p' by using r, α, and β as follows:
x' = r cos (α + β)
y' = r sin (α + β)

Moving, Rotating, and Scaling

97

Then you can use the addition theorem of trigonometric functions1 to get the following:
x' = r (cos α cos β – sin α sin β)
y' = r (sin α cos β + cos α sin β)
Finally, you get the following expressions (Equation 3.3) by assigning Equation 3.2 to the
previous expressions and removing r and α.
Equation 3.3
x' = x cos β – y sin β
y' = x sin β + y cos β
z' = z
So by passing the values of sin β and cos β to the vertex shader and then calculating
Equation 3.3 in the shader, you get the coordinates of the rotated point. To calculate sin β
and cos β, you can use the methods of the JavaScript Math object.
Let’s look at a sample program, RotatedTriangle, which rotates a triangle around the
z-axis, in a counterclockwise direction, by 90 degrees. Figure 3.23 shows RotatedTriangle.

Figure 3.23

1

RotatedTriangle

sin(a ± b) = sin a cos b ∓ cos a sin b
cos(a ± b) = cos a cos b ∓ sin a sin b

98

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Sample Program (RotatedTriangle.js)
Listing 3.5 shows RotatedTriangle.js which, in a similar manner to
TranslatedTriangle.js, modifies the vertex shader to carry out the rotation operation.
The fragment shader is the same as in TranslatedTriangle.js and, as usual, is not shown.
Again, to support the shader modification, several processing steps are added to main() in
the JavaScript program. Additionally, Equation 3.3 is added in the comments from lines 4
to 6 to remind you of the calculation needed.
Listing 3.5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
22
23
24
25
42
43
49
50
51
52
53
54
55

RotatedTriangle.js

// RotatedTriangle.js
// Vertex shader program
var VSHADER_SOURCE =
// x' = x cos b - y sin b
// y' = x sin b + y cos b
Equation 3.3
// z' = z
'attribute vec4 a_Position;\n' +
'uniform float u_CosB, u_SinB;\n' +
'void main() {\n' +
' gl_Position.x = a_Position.x * u_CosB - a_Position.y *u_SinB;\n'+
' gl_Position.y = a_Position.x * u_SinB + a_Position.y * u_CosB;\n'+
' gl_Position.z = a_Position.z;\n' +
' gl_Position.w = 1.0;\n' +
'}\n';
// Fragment shader program
...
// Rotation angle
var ANGLE = 90.0;
function main() {
...
// Set the positions of vertices
var n = initVertexBuffers(gl);
...
// Pass the data required to rotate the shape to the vertex shader
var radian = Math.PI * ANGLE / 180.0; // Convert to radians
var cosB = Math.cos(radian);
var sinB = Math.sin(radian);
var u_CosB = gl.getUniformLocation(gl.program, 'u_CosB');
var u_SinB = gl.getUniformLocation(gl.program, 'u_SinB');
...

Moving, Rotating, and Scaling

99

60
61
62
63
69
70
71
72
73
74
75
76
77

gl.uniform1f(u_CosB, cosB);
gl.uniform1f(u_SinB, sinB);
// Set the color for clearing 
...
// Draw a triangle
gl.drawArrays(gl.TRIANGLES, 0, n);
}

function initVertexBuffers(gl) {
var vertices = new Float32Array([
0.0, 0.5,
-0.5, -0.5,
0.5, -0.5
]);
var n = 3; // The number of vertices
...
105
return n;
106 }

Let’s look at the vertex shader, which is straightforward:
2
3
4
5
6
7
8
9
10
11
12
13
14

// Vertex shader program
var VSHADER_SOURCE =
// x' = x cos b - y sin b
// y' = x sin b + y cos b
// z' = z
'attribute vec4 a_Position;\n' +
'uniform float u_CosB, u_SinB;\n' +
'void main() {\n' +
' gl_Position.x = a_Position.x * u_CosB - a_Position.y * u_SinB;\n'+
' gl_Position.y = a_Position.x * u_SinB + a_Position.y * u_CosB;\n'+
' gl_Position.z = a_Position.z;\n' +
' gl_Position.w = 1.0;\n' +
'}\n';

Because the goal is to rotate the triangle by 90 degrees, the sine and cosine of 90 need to
be calculated. Line 8 defines two uniform variables for receiving these values, which are
calculated in the JavaScript program and then passed to the vertex shader.
You could pass the rotation angle to the vertex shader and then calculate the values of
sine and cosine in the shader. However, because they are identical for all vertices, it is
more efficient to do it once in the JavaScript.
The name of these uniform variables, u_CosB and u_SinB, are defined following the
naming rule used throughout this book. As you will remember, you use the uniform variable because the values of these variables are uniform (unchanging) per vertex.

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As in the previous sample programs, x, y, z, and w are passed in a group to the attribute
variable a_Position in the vertex shader. To apply Equation 3.3 to x, y, and z, you need
to access each component in a_Position separately. You can do this easily using the.
operator, such as a_Position.x, a_Position.y, and a_Position.z (see Figure 3.24 and
Chapter 6).
v e c 4 a _ P o s itio n

x

y

z
a _ P o s itio n .z

w

a _ P o s itio n .x

a _ P o s itio n .y

a _ P o s itio n .w

Figure 3.24

Access methods for each component in a vec4

Handily, you can use the same operator to access each component in gl_Position to
which the vertex coordinate is written, so you can calculate x' = x cos β – y sin β from
Equation 3.3 as shown at line 10:
10

'

gl_Position.x = a_Position.x * u_CosB - a_Position.y * u_SinB;\n'+

Similarly, you can calculate y' as follows:
11

'

gl_Position.y = a_Position.x * u_SinB + a_Position.y * u_CosB;\n'+

According to Equation 3.3, you just need to assign the original z coordinate to z' directly
at line 12. Finally, you need to assign 1.0 to the last component w2:
12
13

'
'

gl_Position.z = a_Position.z;\n' +
gl_Position.w = 1.0;\n' +

Now look at main() in the JavaScript code, which starts from line 25. This code is mostly
the same as in TranslatedTriangle.js. The only difference is passing cos β and sin β to
the vertex shader. To calculate the sine and cosine of β, you can use the JavaScript Math.
sin() and Math.cos() methods. However, these methods expect parameters in radians,
not degrees, so you need to convert from degrees to radians by multiplying the number
of degrees by pi and then dividing by 180. You can utilize Math.PI as the value of pi as
shown at line 50, where the variable ANGLE is defined as 90 (degrees) at line 23:
50

2

var radian = Math.PI * ANGLE / 180.0; // Converts degrees to radians

In this program, you can also write gl_Position.w = a_Position.w; because a_Position.w
is 1.0.

Moving, Rotating, and Scaling

101

Once you have the angle in radians, lines 51 and 52 calculate cos β and sin β, and then
lines 60 and 61 pass them to the uniform variables in the vertex shader:
51
52
53
54
55
60
61

var cosB = Math.cos(radian);
var sinB = Math.sin(radian);
var u_CosB = gl.getUniformLocation(gl.program, 'u_CosB');
var u_SinB = gl.getUniformLocation(gl.program, 'u_SinB');
...
gl.uniform1f(u_CosB, cosB);
gl.uniform1f(u_SinB, sinB);

When you load this program into your browser, you can see the triangle, rotated through
90 degrees, on the screen. If you specify a negative value to ANGLE, you can rotate the
triangle in the opposite direction (clockwise). You can also use the same equation. For
example, to rotate the triangle in the clockwise direction, you can specify –90 instead of
90 at line 23, and Math.cos() and Math.sin() will deal with the remaining tasks for you.
For those of you concerned with speed and efficiency, the approach taken here (using two
uniform variables to pass the values of cos β and sin β) isn’t optimal. To pass the values as
a group, you can define the uniform variable as follows:
uniform vec2 u_CosBSinB;

and then pass the values by:
gl.uniform2f(u_CosBSinB,cosB, sinB);

Then in the vertex shader, you can access them using u_CosBSinB.x and u_CosBSinB.y.

Transformation Matrix: Rotation
For simple transformations, you can use mathematical expressions. However, as your
needs become more complex, you’ll quickly find that applying a series of equations
becomes quite complex. For example a “translation after rotation” as shown in Figure 3.25
can be realized by using Equations 3.1 and 3.3 to find the new mathematical expressions
for the transformation and then implementing them in a vertex shader.
(1)
(2)

x
Figure 3.25

102

Rotate first and then translate a triangle

CHAPTER 3

Drawing and Transforming Triangles

However, it is time consuming to determine the mathematical expressions every time
you need a new set of transformation and then implement them in a vertex shader.
Fortunately, there is another tool in the mathematical toolbox, the transformation
matrix, which is excellent for manipulating computer graphics.
As shown in Figure 3.26, a matrix is a rectangular array of numbers arranged in rows (in
the horizontal direction) and columns (in the vertical direction). This notation makes it
easy to write the calculations explained in the previous sections. The brackets indicate
that these numbers are a group.

⎡8 3 0 ⎤
⎢4 3 6⎥
⎢
⎥
⎢⎣ 3 2 6⎥⎦
Figure 3.26

Example of a matrix

Before explaining the details of how to use a transformation matrix to replace the equations used here, you need to make sure you understand the multiplication of a matrix and
a vector. A vector is an object represented by an n-tuple of numbers, such as the vertex
coordinates (0.0, 0.5, 1.0).
The multiplication of a matrix and a vector can be written as shown in Equation 3.4.
(Although the multiply operator × is often omitted, we explicitly write the operator in this
book for clarity.) Here, our new vector (on the left) is the result of multiplying a matrix
(in the center) by our original vector (on the right). Note that matrix multiplication is
noncommutative. In other words, A × B is not the same as B × A. We discuss this further
in Chapter 6.
Equation 3.4

⎡ x '⎤ ⎡ a b
⎢ y '⎥ = ⎢ d e
⎢ ⎥ ⎢
⎢⎣ z ' ⎥⎦ ⎢⎣ g h

c ⎤ ⎡ x⎤
f ⎥⎥ × ⎢⎢ y ⎥⎥
i ⎥⎦ ⎢⎣ z ⎥⎦

This matrix has three rows and three columns and is called a 3×3 matrix. The rightmost
part of the equation is a vector composed of x, y, and z. (In the case of a multiplication
of a matrix and vector, the vector is written vertically, but it has the same meaning as
when it is written horizontally.) This vector has three elements, so it is called a threedimensional vector. Again, the brackets on both sides of the array of numbers (vector)
are also just notation for recognizing that these numbers are a group.
In this case, x', y', and z' are defined using the elements of the matrix and the vector,
as shown by Equation 3.5. Note that the multiplication of a matrix and vector can be

Moving, Rotating, and Scaling

103

defined only if the number of columns in a matrix matches the number of rows in a
vector.
Equation 3.5
x' = ax + by + cz
y' = dx + ey + fz
z' = gx + hy + iz
Now, to understand how to use a matrix instead of our original equations, let’s compare
the matrix equations and Equation 3.3 (shown again as Equation 3.6).
Equation 3.6
x' = x cos β – y sin β
y' = x sin β + y cos β
z' = z
For example, compare the equation for x':
x' = ax + by + cz
x' = x cos β – y sin β
In this case, if you set a = cos β, b = -sin β, and c = 0, the equations become the same.
Similarly, let us compare the equation for y':
y' = dx + ey + fz
y' = x sin β + y cos β
In this case, if you set d = sin β, e = cos β, and f = 0, you get the same equation. The last
equation about z' is easy. If you set g = 0, h = 0, and i = 1, you get the same equation.
Then, by assigning these results to Equation 3.4, you get Equation 3.7.
Equation 3.7

⎡ x ' ⎤ ⎡ cos β
⎥ ⎢
⎢
⎢ y ' ⎥ = ⎢ sin β
⎢ z' ⎥ ⎢ 0
⎦ ⎣
⎣

− sin β
cos β
0

0 ⎤ ⎡ x ⎤
⎥ ⎢
⎥
0 ⎥×⎢ y ⎥
1 ⎥⎦ ⎢⎣ z ⎥⎦

This matrix is called a transformation matrix because it “transforms” the right-side
vector (x, y, z) to the left-side vector (x', y', z'). The transformation matrix representing a
rotation is called a rotation matrix.
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You can see that the elements of the matrix in Equation 3.7 are an array of coefficients in
Equation 3.6. Once you become accustomed to matrix notation, it is easier to write and
use matrices than to have to deal with a set of transformation equations.
As you would expect, because matrices are used so often in 3DCG, multiplication of a
matrix and a vector is easy to implement in shaders. However, before exploring how, let’s
quickly look at other types of transformation matrices, and then we will start to use them
in shaders.

Transformation Matrix: Translation
Obviously, if we can use a transformation matrix to represent a rotation, we should be
able to use it for other types of transformation, such as translation. For example, let us
compare the equation for x' in Equation 3.1 to that in Equation 3.5 as follows:
x' = ax + by + cz

- - - from Equation (3.5)

x' = x + Tx

- - - from Equation (3.1)

Here, the second equation has the constant term Tx, but the first one does not, meaning
that you cannot deal with the second one by using the 3×3 matrix of the first equation.
To solve this problem, you can use a 4×4 matrix and the fourth components of the coordinate, which are set to 1 to introduce the constant terms. That is to say, we assume that
the coordinates of point p are (x, y, z, 1), and the coordinates of the translated point p (p')
are (x', y', z', 1). This gives us Equation 3.8.
Equation 3.8

⎡
⎢
⎢
⎢
⎢
⎣

x' ⎤ ⎡
⎥ ⎢
y' ⎥ ⎢
=
z' ⎥ ⎢
⎥ ⎢
1 ⎦ ⎢⎣

a
e

b
f

i

j

m

n

c d ⎤ ⎡ x ⎤
⎥
g h ⎥ ⎢ y ⎥
⎥
×⎢
k l ⎥ ⎢ z ⎥
⎥ ⎢
⎥
o p ⎥ ⎣ 1 ⎦
⎦

This multiplication is defined as follows:
Equation 3.9
x' = ax + by + cz +d
y' = ex + fy + gz + h
z' = ix + jy + kz + l
1 = mx + ny + oz+ p

Moving, Rotating, and Scaling

105

From the equation1 = mx + ny + oz + p, it is easy to find that the coefficients are m = 0,
n = 0, o = 0, and p = 1. In addition, these equations have the constant terms d, h, and l,
which look helpful to deal with Equation 3.1 because it also has constant terms. Let us
compare Equation 3.9 and Equation 3.1 (translation), which is reproduced again:
x' = x + Tx
y' = y + Ty
z' = z + Tz
When you compare the x' component of both equations, you can see that a=1, b=0, c=0,
and d=Tx. Similarly, when comparing y' from both equations, you find e = 0, f = 1, g = 0,
and h = Ty; when comparing z' you see i=0, j=0, k=1, and l=Tz. You can use these results
to write a matrix that represents a translation, called a translation matrix, as shown in
Equation 3.10.
Equation 3.10
⎡
⎢
⎢
⎢
⎢
⎣

x' ⎤ ⎡
⎥ ⎢
y' ⎥ ⎢
=
z' ⎥ ⎢
⎥ ⎢
1 ⎦ ⎣

1 0 0 Tx ⎤ ⎡
⎥ ⎢
0 1 0 Ty ⎥ ⎢
×
0 0 1 Tz ⎥ ⎢
⎥ ⎢
0 0 0 1 ⎦ ⎣

x ⎤
⎥
y ⎥
z ⎥
⎥
1 ⎦

Rotation Matrix, Again
At this stage you have successfully created a rotation and a translation matrix, which are
equivalent to the two equations you used in the example programs earlier. The final step
is to combine these two matrices; however, the rotation matrix (3×3 matrix) and transformation matrix (4×4 matrix) have different numbers of elements. Unfortunately, you
cannot combine matrices of different sizes, so you need a mechanism to make them the
same size.
To do that, you need to change the rotation matrix (3×3 matrix) into a 4×4 matrix. This is
straightforward and requires you to find the coefficient of each equation in Equation 3.9
by comparing it with Equation 3.3. The following shows both equations:
x' = x cos β – y sin β
y' = x sin β + y cos β
z' = z

106

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Drawing and Transforming Triangles

x' = ax + by + cz + d
y' = ex + fy + gz + h
z' = ix + iy + kz + l
1 = mx + ny + oz + p
For example, when you compare x' = x cos β – y sin β with x' = ax + by + cz +d, you find
a = cos β, b = –sin β, c = 0, and d = 0. In the same way, after comparing in terms of y and
z, you get the rotation matrix shown in Equation 3.11:
Equation 3.11
⎡
⎢
⎢
⎢
⎢
⎣

x ' ⎤ ⎡ cos β
⎥ ⎢
y ' ⎥ ⎢ sin β
=
z' ⎥ ⎢ 0
⎥ ⎢
1 ⎦ ⎢⎣ 0

− sin β
cos β
0
0

0 0 ⎤ ⎡
⎥ ⎢
0 0 ⎥ ⎢
×
1 0 ⎥ ⎢
⎥ ⎢
0 1 ⎥⎦ ⎣

x ⎤
⎥
y ⎥
z ⎥
⎥
1 ⎦

This allows you to represent both a rotation matrix and translation matrix in the same
4×4 matrix, achieving the original goal!

Sample Program (RotatedTriangle_Matrix.js)
Having constructed a 4×4 rotation matrix, let’s go ahead and use this matrix in a WebGL
program by rewriting the sample program RotatedTriangle, which rotates a triangle
90 degrees around the z-axis in a counterclockwise direction, using the rotation matrix.
Listing 3.6 shows RotatedTriangle_Matrix.js, whose output will be the same as Figure
3.23 shown earlier.
Listing 3.6
1
2
3
4
5
6
7
8
9
10
16
17

RotatedTriangle_Matrix.js

// RotatedTriangle_Matrix.js
// Vertex shader program
var VSHADER_SOURCE =
'attribute vec4 a_Position;\n' +
'uniform mat4 u_xformMatrix;\n' +
'void main() {\n' +
' gl_Position = u_xformMatrix * a_Position;\n' +
'}\n';
// Fragment shader program
...
// Rotation angle
var ANGLE = 90.0;

Moving, Rotating, and Scaling

107

18
19
36
37
43
44
45
46
47
48
49
50
51
52
53
54
55
56
61
62
63
69
70
71
72
73
74
75
76
77
105
106

function main() {
...
// Set the positions of vertices
var n = initVertexBuffers(gl);
...
// Create a rotation matrix
var radian = Math.PI * ANGLE / 180.0; // Convert to radians
var cosB = Math.cos(radian), sinB = Math.sin(radian);
// Note: WebGL is column major order
var xformMatrix = new Float32Array([
cosB, sinB, 0.0, 0.0,
-sinB, cosB, 0.0, 0.0,
0.0, 0.0, 1.0, 0.0,
0.0, 0.0, 0.0, 1.0
]);
// Pass the rotation matrix to the vertex shader
var u_xformMatrix = gl.getUniformLocation(gl.program, 'u_xformMatrix');
...
gl.uniformMatrix4fv(u_xformMatrix, false, xformMatrix);
// Set the color for clearing 
...
// Draw a triangle
gl.drawArrays(gl.TRIANGLES, 0, n);
}
function initVertexBuffers(gl) {
var vertices = new Float32Array([
0.0, 0.5,
-0.5, -0.5,
0.5, -0.5
]);
var n = 3; // Number of vertices
...
return n;
}

First, let us examine the vertex shader:
2
3
4
5

// Vertex shader program
var VSHADER_SOURCE =
'attribute vec4 a_Position;\n' +
'uniform mat4 u_xformMatrix;\n' +

108

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6
7
8

'void main() {\n' +
' gl_Position = u_xformMatrix * a_Position;\n' +
'}\n';

At line 7, u_xformMatrix, containing the rotation matrix described in Equation 3.11, and
a_Position, containing the vertex coordinates (this is the right-side vector in Equation
3.11), are multiplied, literally implementing Equation 3.11.
In the sample program TranslatedTriangle, you were able to implement the addition of
two vectors in one line (gl_Position = a_Position + u_Translation). In the same way, a
multiplication of a matrix and vector can be written in one line in GLSL ES. This is convenient, allowing the calculation of the four equations (Equation 3.9) in one line. Again,
this shows how GLSL ES has been designed specifically for 3D computer graphics by
supporting powerful operations like this.
Because the transformation matrix is a 4×4 matrix and GLSL ES requires the data type for
all variables, line 5 declares u_xformMatrix as type mat4. As you would expect, mat4 is a
data type specifically for holding a 4×4 matrix.
Within the main JavaScript program, the rest of the changes just calculate the rotation
matrix from Equation 3.11 and then pass it to u_xformMatrix. This part starts from
line 44:
43
44
45
46
47
48
49
50
51
52
53
54
55
61

// Create a rotation matrix
var radian = Math.PI * ANGLE / 180.0; // Convert to radians
var cosB = Math.cos(radian), sinB = Math.sin(radian);
// Note: WebGL is column major order
var xformMatrix = new Float32Array([
cosB, sinB, 0.0, 0.0,
-sinB, cosB, 0.0, 0.0,
0.0, 0.0, 1.0, 0.0,
0.0, 0.0, 0.0, 1.0
]);
// Pass the rotation matrix to the vertex shader
...
gl.uniformMatrix4fv(u_xformMatrix, false, xformMatrix);

Lines 44 and 45 calculate the values of cosine and sine, which are required in the rotation matrix. Then line 48 creates the matrix xformMatrix using a Float32Array. Unlike
GLSL ES, because JavaScript does not have a dedicated object for representing a matrix,
you need to use the Float32Array. One question that arises is in which order you should
store the elements of the matrix (which is arranged in rows and columns) in the elements
of the array (which is arranged in a line). There are two possible orders: row major order
and column major order (see Figure 3.27).

Moving, Rotating, and Scaling

109

a
e
i
m

b
f
j
n

c
g
k
o

a
e
i
m

d
h
l
p

r o w m a jo r
Figure 3.27

b
f
j
n

c
g
k
o

d
h
l
p

c o lu mn m a jo r
Row major order and column major order

WebGL, just like OpenGL, requires you to store the elements of a matrix in the elements
of an array in column major order. So, for example, the matrix shown in Figure 3.27
is stored in an array as follows: [a, e, i, m, b, f, j, n, c, g, k, o, d, h, l, p]. In the sample
program, the rotation matrix is stored in the Float32Array in this order in lines 49 to 52.
The array created is then passed to the uniform variable u_xformMatrix by using
gl.uniformMatrix4fv() at line 61. Note that the last letter of this method name is v,
which indicates that the method can pass multiple data values to the variable.

gl.uniformMatrix4fv(location, transpose, array)
Assign the 4×4 matrix specified by array to the uniform variable specified by location.
Parameters

location

Specifies the storage location of the uniform variable.

Transpose

Must be false in WebGL.3

array

Specifies an array containing a 4×4 matrix in column
major order (typed array).

Return value

None

Errors

INVALID_OPERATION

There is no current program object.

INVALID_VALUE

transpose is not false, or the length of array is less
than 16.

If you load and run the sample program in your browser, you’ll see the rotated triangle.
Congratulations! You have successfully learned how to use a transformation matrix to
rotate a triangle.

3

This parameter specifies whether to transpose the matrix or not. The transpose operation, which
exchanges the column and row elements of the matrix (see Chapter 7), is not supported by WebGL’s
implementation of this method and must always be set to false.

110

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Drawing and Transforming Triangles

Reusing the Same Approach for Translation
Now, as you have seen with Equations 3.10 and 3.11, you can represent both a translation
and a rotation using the same type of 4×4 matrix. Both equations use the matrices in the
form  =  * . This
is coded in the vertex shader as follows:
7

'

gl_Position = u_xformMatrix * a_Position;\n' +

This means that if you change the elements of the array xformMatrix from those of a
rotation matrix to those of a translation matrix, you will be able to apply the translation
matrix to the triangle to achieve the same result as shown earlier but which used an equation (Figure 3.18).
To do that, change line 17 in RotatedTriangle_Matrix.js using the translation distances
from the previous example:
17

varTx = 0.5, Ty = 0.5, Tz = 0.0;

You need to rewrite the code for creating the matrix, remembering that you need to store
the elements of the matrix in column major order. Let’s keep the same name for the array
variable, xformMatrix, even though it’s now being used to hold a translation matrix,
because it reinforces the fact that we are using essentially the same code. Finally, you are
not using the variable ANGLE, so lines 43 to 45 are commented out:
43
44
45
46
47
48
49
50
51
52
53

// Create a rotation matrix
// var radian = Math.PI * ANGLE / 180.0; // Convert to radians
// var cosB = Math.cos(radian), sinB = Math.sin(radian);
// Note: WebGL is column major order
var xformMatrix = new Float32Array([
1.0, 0.0, 0.0, 0.0,
0.0, 1.0, 0.0, 0.0,
0.0, 0.0, 1.0, 0.0,
Tx, Ty, Tz, 1.0
]);

Once you’ve made the changes, run the modified program, and you will see the same
output as shown in Figure 3.18. By using a transformation matrix, you can apply various
transformations using the same vertex shader. This is why the transformation matrix is
such a convenient and powerful tool for 3D graphics, and it’s why we’ve covered it in
detail in this chapter.

Transformation Matrix: Scaling
Finally, let’s define the transformation matrix for scaling using the same assumption that
the original point is p and the point after scaling is p'.

Moving, Rotating, and Scaling

111

y

p(x , y , z)
x
p ' ( x ' , y ' , z' )
Figure 3.28

A scaling transformation

Assuming the scaling factor for the x-axis, y-axis, and z-axis is Sx, Sy, and Sz respectively,
you obtain the following equations:
x' = Sx × x
y' = Sy × y
z' = Sz × z
The following transformation matrix can be obtained by comparing these equations with
Equation 3.9.
⎡
⎢
⎢
⎢
⎢
⎣

x' ⎤ ⎡
⎥ ⎢
y' ⎥ ⎢
=
z' ⎥ ⎢
⎥ ⎢
1 ⎦ ⎣

Sx 0
0 Sy
0
0

0
0

0 ⎤ ⎡ x ⎤
⎥
⎥ ⎢
0 ⎥ ⎢ y ⎥
×
Sz 0 ⎥ ⎢ z ⎥
⎥
⎥ ⎢
0 1 ⎦ ⎣ 1 ⎦
0
0

As with the previous example, if you store this matrix in xformMatrix, you can scale the
triangle by using the same vertex shader you used in RotatedTriangle_Matrix.js. For
example, the following sample program will scale the triangle by a factor of 1.5 in a vertical direction, as shown in Figure 3.29:
17
47
48
49
50
51
52
53

112

varSx = 1.0, Sy = 1.5, Sz = 1.0;
...
// Note: WebGL is column major order
var xformMatrix = new Float32Array([
Sx, 0.0, 0.0, 0.0,
0.0, Sy, 0.0, 0.0,
0.0, 0.0, Sz, 0.0,
0.0, 0.0, 0.0, 1.0
]

CHAPTER 3

Drawing and Transforming Triangles

Figure 3.29

Triangle scaled in a vertical direction

Note that if you specify 0.0 to Sx, Sy, or Sz, the scaled size will be 0.0. If you want to keep
the original size, specify 1.0 as the scaling factor.

Summary
In this chapter, you explored the process of passing multiple items of information about
vertices to a vertex shader, the different types of shapes available to be drawn using that
information, and the process of transforming those shapes. The shapes dealt with in this
chapter changed from a point to a triangle, but the method of using shaders remained the
same, as in the examples in the previous chapter. You were also introduced to matrices
and learned how to use transformation matrices to apply translation, rotation, or scaling
to 2D shapes. Although it’s a little complicated, you should now have a good understanding of the math behind calculating the individual transformation matrices.
In the next chapter, you’ll explore more complex transformations but will use a handy
library to hide the details, allowing you to focus on the higher-level tasks.

Summary

113

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Chapter 4

More Transformations and Basic Animation

Chapter 3, “Drawing and Transforming Triangles,” explained how to use the buffer object to
draw more complex shapes and how to transform them using simple equations. Because this
approach is cumbersome for multiple transformations, you were then introduced to matrices and
their use to simplify transformation operations. In this chapter, you explore further transformations and begin to combine transformations into animations. In particular, you will

• Be introduced to a matrix transformation library that hides the mathematical details of
matrix operations

• Use the library to quickly and easily combine multiple transformations
• Explore animation and how the library helps you animate simple shapes
These techniques provide the basics to construct complex WebGL programs and will be used in
the sample programs in the following chapters.

Translate and Then Rotate
As you saw in Chapter 3, although transformations such as translation, rotation, and scaling can
be represented as a 4×4 matrix, it is time consuming to specify each matrix by hand whenever
you write WebGL programs. To simply the task, most WebGL developers use a convenient library
that automates creating these matrices and hides the details. There are several public matrix
libraries for WebGL, but in this section, you’ll use a library created for this book and will see how
the library can be used to combine multiple transformations to achieve results such as “translate
and then rotate a triangle.”

Transformation Matrix Library: cuon-matrix.js
In OpenGL, the parent specification of WebGL, there’s no need to specify by hand each
element of the transformation matrix when you use it. Instead, OpenGL supports a set
of handy functions that make it easy to construct the matrices. For example, glTranslatef(), when called with translation distances for the x-, y-, and z-axes as arguments,
will create a translation matrix internally and then set up the matrix needed to translate a
shape (see Figure 4.1).

glTranslatef(5, 80, 30);

Figure 4.1

1
0
0
0

0
1
0
0

0
0
1
0

5
80
30
1

An example of glTranslatef() in OpenGL

Unfortunately, WebGL doesn’t provide these functions, so if you wanted to use this
approach you’d need to look for or write them yourself. Because these functions are
useful, we’ve created a JavaScript library for you, cuon-matrix.js, which allows you to
create transformation matrices in a similar way to those specified in OpenGL. Although
this library was written specifically for this book, it is general purpose, and you are free to
use it in your own applications.
As you saw in Chapter 3, JavaScript provides the sine and cosine function as methods of
the Math object. In the same way, the functions for creating transformation matrices are
provided as methods of the Matrix4 object defined in cuon-matrix.js.
Matrix4 is a new object, defined by the library, and as the name suggests, deals with 4×4
matrices. Internally, these are represented using a typed array, Float32Array, which you

saw in Chapter 3.
To get a feel for the library, let’s rewrite RotatedTriangle_Matrix using the Matrix4 object
and its methods. The program for this example is called RotatedTriangle_Matrix4.
Because the Matrix4 object is defined in the library cuon-matrix.js, you need to load it
into your HTML file before using the object. To do this, you simply use the 
src="../lib/cuon-utils.js">
src="../lib/cuon-matrix.js">
src="RotatedTriangle_Matrix4.js">

Once loaded, let’s examine how to use it by comparing the previous RotatedTriangle_
Matrix.js with RotatedTriangle_Matrix4.js, which uses the new Matrix4 object.

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Sample Program (RotatedTriangle_Matrix4.js)
This sample program is almost the same as RotatedTriangle_Matrix.js used in Chapter
3 except for the new steps that create the transformation matrix and pass it to u_xformMatrix in the vertex shader.
In RotatedTriangle_Matrix.js, you created the transformation matrix as follows:
1

// RotatedTriangle_Matrix.js
...
43
// Create a rotation matrix
44
var radian = Math.PI * ANGLE / 180.0; // Convert to radians
45
varcosB = Math.cos(radian), sinB = Math.sin(radian);
46
47
// Note: WebGL is column major order
48
var xformMatrix = new Float32Array([
49
cosB, sinB, 0.0, 0.0,
50
-sinB, cosB, 0.0, 0.0,
51
0.0, 0.0, 1.0, 0.0,
52
0.0, 0.0, 0.0, 1.0
53
]);
...
61
gl.uniformMatrix4fv(u_xformMatrix, false, xformMatrix);

In this sample program, you need to rewrite this part using a Matrix4 object and utilize its
method setRotate() to calculate a rotation matrix. The following code snippet shows the
rewritten part from RotatedTriangle_Matrix4.js:
1
47
48
49
50
56
57

// RotatedTriangle_Matrix4.js
...
// Create Matrix4 object for a rotation matrix
var xformMatrix = new Matrix4();
// Set the rotation matrix to xformMatrix
xformMatrix.setRotate(ANGLE, 0, 0, 1);
...
// Pass the rotation matrix to the vertex shader
gl.uniformMatrix4fv(u_xformMatrix, false, xformMatrix.elements);

You can see that the basic processing flow to create a matrix (line 48 and 50) and then
pass it to the uniform variable is the same as in the previous sample. A Matrix4 object is
created using the new operator in the same way that an Array or Date object is created in
JavaScript. Line 48 creates the Matrix4 object, and then line 50 uses setRotate() to calculate the rotation matrix and write it to the xformMatrix object.
The setRotate() method takes four parameters: a rotation angle (specified in degrees, not
radians) and the rotation axis around which the rotation will take place. The rotation axis,
specified by x, y, and z, and associated direction are defined by the line drawn from

Translate and Then Rotate

117

(0, 0, 0) to (x, y, z). As you will remember from Chapter 3 (Figure 3.21), the angle is positive if the rotation is performed in the clockwise direction around the rotation axis. In this
example, because the rotation is performed around the z-axis, the axis is set to (0, 0, 1):
50

xformMatrix.setRotate(ANGLE, 0, 0, 1);

Similarly, for the x-axis, you can specify x = 1, y =0, and z = 0 and for the y-axis, x = 0,
y = 1, and z = 0. Once you have set up the rotation matrix in the variable xformMatrix,
all you need to do is to pass the matrix to the vertex shader using the same method gl.
uniformMatrix4fv() as before. Note that you cannot pass a Matrix4 object directly as
the last argument of this method because a typed array is required for the parameter.
However, you can retrieve the elements of the Matrix4 object as a typed array by using its
property elements, which you can then pass as shown at line 57:
57

gl.uniformMatrix4fv(u_xformMatrix, false, xformMatrix.elements);

The Matrix4 object supports the methods and properties shown in Table 4.1.
Table 4.1

The Methods and Properties Supported by Matrix4

Methods and Properties

Description

Matrix4.setIdentity()

Initialize a matrix (to the identity matrix*).

Matrix4.setTranslate(x, y,
z)

Set Matrix4 to the translation matrix, which translates x
units in the direction of the x-axis, y units in the direction of
the y-axis, and z units in the direction of the z-axis.

Matrix4.setRotate(angle,
x, y, z)

Set Matrix4 to the rotation matrix, which rotates angle
degrees around the rotation axis (x, y, z). The (x, y, z) coordinates do not need to be normalized. (See Chapter 8,
“Lighting Objects.”)

Matrix4.setScale(x, y, z)

Set Matrix4 to the scaling matrix with scaling factors x, y,
and z.

Matrix4.translate (x, y,
z)

Multiply the matrix stored in Matrix4 by the translation
matrix, which translates x units in the direction of the x-axis,
y units in the direction of the y-axis, and z units in the direction of the z-axis, storing the result back into Matrix4.

Matrix4.rotate(angle, x,
y, z)

Multiply the matrix stored in Matrix4 by the rotation matrix,
which rotates angle degrees around the rotation axis (x, y,
z), storing the results back into Matrix4. The (x, y, z) coordinates do not need to be normalized. (See Chapter 8.)

Matrix4.scale(x, y, z)

Multiply the matrix stored in Matrix4 by the scaling matrix,
with scaling factors x, y, and z, storing the results back into
Matrix4.

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Methods and Properties

Description

Matrix4.set(m)

Set the matrix m to Matrix4. m must be a Matrix4 object.

Matrix4.elements

The typed array (Float32Array) containing the elements of
the matrix stored in Matrix4.

* The identity matrix is a matrix that behaves for matrix multiplication like the scalar value 1 does for
scalar multiplication. Multiplying by the identity matrix has no effect on the other matrix. The identity
matrix has 1.0 in its diagonal elements.

As can be seen from the table, there are two types of methods supported by Matrix4: those
whose names include set, and those that don’t. The set methods calculate the transformation matrix using their parameters and then set or write the resulting matrix to the
Matrix4 object. In contrast, the methods without set multiply the matrix already stored
in the Matrix4 object by the matrix calculated using the parameters and then store the
result back in the Matrix4 object.
As you can see from the table, these methods are both powerful and flexible. More importantly, it makes it easy to quickly change transformations. For example, if you wanted to
translate the triangle instead of rotating, you could just rewrite line 50 as follows:
50

xformMatrix.setTranslate( 0.5, 0.5, 0.0);

The example uses the variable name xformMatrix to represent a generic transformation.
Obviously, you can use an appropriate variable name for the transformations within your
applications (such as rotMatrix, as was used in Chapter 3).

Combining Multiple Transformation
Now that you are familiar with the basics of the Matrix4 object, let’s see how it can be
used to combine two transformations: a translation followed by a rotation. The sample
program RotatedTranslatedTriangle does just that, resulting in Figure 4.2. Note that
the example uses a smaller triangle so that you can easily understand the effect of the
transformation.

Translate and Then Rotate

119

Figure 4.2

RotatedTranslatedTriangle

Obviously, this example consists of the following two transformations, shown in
Figure 4.3:
1. Translate the triangle along the x-axis.
2. Rotate the triangle translated by (1).

(2)

y
(1)

x
Figure 4.3

The triangle translated and then rotated

Based on the explanations so far, we can write the equation of the translated triangle from
(1) as follows:
Equation 4.1

〈" translated " coordinates〉 =
〈translation matrix 〉 × 〈original coordinates〉
Then you just need to rotate the < " translated " coordinates > as follows:
Equation 4.2

〈" translated and then rotated " coordinates〉 =
〈rotation matrix 〉 × 〈translated coordinates〉
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You can calculate these equations separately, but they can be combined by substituting
Equation 4.1 into Equation 4.2.
Equation 4.3
〈”translated and then rotated ” coordinates 〉 =
〈rotation matrix 〉 × ( 〈translation matrix 〉 × 〈original coordinates 〉 )
Where
〈rotation matrix 〉 × ( 〈translation matrix 〉 × 〈original coordinates 〉 )
is equal to

( 〈rotation matrix 〉 × 〈translation matrix 〉 ) × 〈original

coordinates 〉

This final step, calculating 〈rotation matrix 〉 × 〈translation matrix 〉 , can be carried out in a
JavaScript program and the result passed to the vertex shader. The combination of multiple transformations like this is called model transformation (or modeling transformation), and the matrix that performs model transformation is called the model matrix.
As a refresher, let’s just look at the multiplication of matrices, which is defined as follows:

⎡ a00
⎢
A = ⎢ a10
⎢ a
⎢⎣ 20

a01
a11
a21

⎡ b00
a02 ⎤
⎥
⎢
a12 ⎥ , B = ⎢ b10
⎢ b
a22 ⎥⎥
⎢⎣ 20
⎦

b01 b02 ⎤
⎥
b11 b12 ⎥
b21 b22 ⎥⎥
⎦

Assuming two 3×3 matrices, A and B as shown, the product of A and B is defined as
follows:
Equation 4.4

⎡ a ×b +a ×b +a ×b
01
10
02
20
⎢ 00 00
⎢ a10 × b00 + a11 × b10 + a12 × b20
⎢
⎢⎣ a20 × b00 + a21 × b10 + a22 × b20

a00 × b01 + a01 × b11 + a02 × b21
a10 × b01 + a11 × b11 + a12 × b21
a20 × b01 + a21 × b11 + a22 × b21

a00 × b02 + a01 × b12 + a02 × b22 ⎤
⎥
a10 × b02 + a11 × b12 + a12 × b22 ⎥
⎥
a20 × b02 + a21 × b12 + a22 × b22 ⎥
⎦

We use 3×3 matrices in the example, but the approach scales to the more usual 4×4 matrices. However, note that the multiplication order of matrices is important. The result of
A * B is not equal to that of B * A.
As you would expect, cuon-matrix.js supports a method to carry out matrix multiplication on Matrix4 objects. Let’s look at how to use that method to combine two matrices to
support a translation followed by a rotation.

Sample Program (RotatedTranslatedTriangle.js)
Listing 4.2 shows RotatedTranslatedTriangle.js. The vertex shader and fragment shader
are the same as in RotatedTriangle_Matrix4.js in the previous section except that the
name of the uniform variable is changed from u_xformMatrix to u_ModelMatrix.
Translate and Then Rotate

121

Listing 4.2
1
2
3
4
5
6
7
8
9
16
33
34
40
41
42
43
44
45
46
47

// RotatedTranslatedTriangle.js
// Vertex shader program
var VSHADER_SOURCE =
'attribute vec4 a_Position;\n' +
'uniform mat4 u_ModelMatrix;\n' +
'void main() {\n' +
' gl_Position = u_ModelMatrix * a_Position;\n' +
'}\n';
// Fragment shader program
...
function main() {
...
// Set the positions of vertices
var n = initVertexBuffers(gl);
...
// Create Matrix4 object for model transformation
var modelMatrix = new Matrix4();
// Calculate a model matrix
var ANGLE = 60.0; // Rotation angle
varTx = 0.5; // Translation distance
modelMatrix.setRotate(ANGLE, 0, 0, 1); // Set rotation matrix
modelMatrix.translate(Tx, 0, 0); // Multiply modelMatrix by the calculated
➥translation matrix

48
49
50

// Pass the model matrix to the vertex shader
var u_ModelMatrix = gl.getUniformLocation(gl.program, ' u_ModelMatrix');
...
gl.uniformMatrix4fv(u_ModelMatrix, false, modelMatrix.elements);
...
// Draw a triangle
gl.drawArrays(gl.TRIANGLES, 0, n);

56
63
64
65
66
67
68
69
70
71
99
100

122

RotatedTranslatedTriangle.js

}
function initVertexBuffers(gl) {
var vertices = new Float32Array([
0.0, 0.3,
-0.3, -0.3,
0.3, -0.3
]);
var n = 3; // The number of vertices
...
return n;
}

CHAPTER 4

More Transformations and Basic Animation

The key lines in this listing are lines 46 and 47, which calculate
< rotation matrix > × < translation matrix > :
46
47

modelMatrix.setRotate(ANGLE, 0, 0, 1); // Set rotation matrix
modelMatrix.translate(Tx, 0, 0); // Multiply modelMatrix by the calculated
➥translation matrix

Because line 46 uses a method with set (setRotate()), the rotation matrix that is calculated using the parameters is written to the variable modelMatrix. The next line, 47,
uses a method without set (translate()), which, as explained earlier, calculates the
translation matrix using the parameters and then multiplies the matrix in modelMatrix by the newly calculated translation matrix and writes the result back into modelMatrix. So, if modelMatrix already contains a rotation matrix, this method calculates
〈rotation matrix 〉 × 〈translation matrix 〉 and stores the result back into modelMatrix.
You may have noticed that the order of “translate first and then rotate” is the opposite of
the order of the matrices in the calculation 〈rotation matrix 〉 × 〈translation matrix 〉 . As shown in
Equation 4.3, this is because the transformation matrix is multiplied by the original vertex
coordinates of the triangle.
The result of this calculation is passed to u_ModelMatrix in the vertex shader at line
56, and then the drawing operation (line 64) is the same as usual. If you now load this
program into your browser, you can see a red triangle, which has been translated and then
rotated.

Experimenting with the Sample Program
Let’s rewrite the sample program to first rotate the triangle and then translate it. This
simply requires you to exchange the order of the rotation and translation. In this
case, you should note that the translation is performed first by using the set method,
setTranslate():
46
47

modelMatrix.setTranslate(Tx, 0, 0);
modelMatrix.rotate(ANGLE, 0, 0, 1);

Figure 4.4 shows this sample program.

Translate and Then Rotate

123

Figure 4.4

A triangle “rotated first and then translated”

As you can see, by changing the order of a rotation and translation, you get a different
result. This becomes obvious when you examine Figure 4.5.

(2)

y
(1)

translate first and then rotate

x
y

(1)
(2)

rotate first and then translate

x
Figure 4.5

The order of transformations will show different results

That concludes the initial explanation of the use of methods defined in cuon-matrix.js
to create transformation matrices. You’ll be using them throughout the rest of this book,
so you’ll have plenty of chance to study them further.

Animation
So far, this chapter has explained how to transform shapes and use the matrix library to
carry out transformation operations. You now have enough knowledge of WebGL to start
on the next step of applying this knowledge to animate shapes.

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Let’s start by constructing a sample program, RotatingTriangle, which continually rotates
a triangle at a constant rotation speed (45 degrees/second). Figure 4.6 shows multiple overlaid screenshots of RotatingTriangle so that you can see the rotation.

Figure 4.6

Multiple overlaid screenshots of RotatingTriangle

The Basics of Animation
To animate a rotating triangle, you simply need to redraw the triangle at a slightly different angle each time it draws.
Figure 4.7 shows individual triangles that are drawn at times t0, t1, t2, t3, and t4. Each
triangle is a still image, but you can see that each has a slightly different rotation angle.
When you see a series of these triangles sequentially, your mind interpolates the changes
between them and then puts them together as a smooth flow of animation, just like a flip
book. Of course, you need to clear the previous triangle before drawing a new one. (This is
why you must call gl.clear() before drawing something.) You can apply this animation
method to both 2D shapes and 3D objects.

t0

t1

t2

t3

t4
time

Figure 4.7

Draw a slightly different triangle for each drawing

Animation

125

Achieving animation in this way requires two key mechanisms:
Mechanism 1: Repeatedly calls a function to draw a triangle at times t0, t1, t2, t3, and
so on.
Mechanism 2: Clears the previous triangle and then draws a new one with the specified
angle each time the function is called.
The second mechanism is just a simple application of the knowledge you’ve learned so far.
However, the first mechanism is new, so let’s take it step by step by examining the sample
program.

Sample Program (RotatingTriangle.js)
Listing 4.3 shows RotatingTriangle.js. The vertex shader and fragment shader are the
same as in the previous sample program. However, the vertex shader is listed to show the
multiplication of a matrix and vertex coordinates.
The following three points differ from the previous sample program:

• Because the program needs to draw a triangle repeatedly, it’s been modified to
specify the clear color at line 44, not just before the drawing operation. Remember,
the color stays in the WebGL system until it’s overwritten.

• The actual mechanism [Mechanism 1] to repeatedly call a drawing function has been
added (lines 59 to 64).

• [Mechanism 2] The operations to clear and draw a triangle were defined as a function (draw() at line 102).
These differences are highlighted in Listing 4.3 (lines 1 to 3). Let’s look at them in more
detail.
Listing 4.3
1
2
3
4
5
6
7
8
9
16
17
18

126

RotatingTriangle.js

// RotatingTriangle.js
// Vertex shader program
var VSHADER_SOURCE =
'attribute vec4 a_Position;\n' +
'uniform mat4 u_ModelMatrix;\n' +
'void main() {\n' +
' gl_Position = u_ModelMatrix * a_Position;\n' +
'}\n';
// Fragment shader program
...
// Rotation angle (degrees/second)
var ANGLE_STEP = 45.0;

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More Transformations and Basic Animation

19
36
37
43
44
45
46
47
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71

function main() {
...
// Set the positions of vertices
var n = initVertexBuffers(gl);
...
// Set the color for clearing 
gl.clearColor(0.0, 0.0, 0.0, 1.0);

<- (1)

// Get the storage location of u_ModelMatrix variable
var u_ModelMatrix = gl.getUniformLocation(gl.program, ' u_ModelMatrix');
...
// Current rotation angle of a triangle
varcurrentAngle = 0.0;
// Matrix4 object for model transformation
var modelMatrix = new Matrix4();
// Start to draw a triangle
var tick = function() {
currentAngle = animate(currentAngle);// Update the rotation angle
draw(gl, n, currentAngle, modelMatrix, u_ModelMatrix);
requestAnimationFrame(tick);// Request that the browser calls tick
};
tick();
}
function initVertexBuffers(gl) {
var vertices = new Float32Array ([
0.0, 0.5,
-0.5, -0.5,
0.5, -0.5
]);
var n = 3;
// The number of vertices
...
return n;
}

96
97
98
99 function draw(gl,n, currentAngle, modelMatrix, u_ModelMatrix){
100
// Set up rotation matrix
101
modelMatrix.setRotate(currentAngle, 0, 0, 1);
102
103
104
105
106
107
108
109

<- (2)

<-(3)

// Pass the rotation matrix to the vertex shader
gl.uniformMatrix4fv( u_ModelMatrix, false, modelMatrix.elements);
// Clear 
gl.clear(gl.COLOR_BUFFER_BIT);
// Draw a triangle

Animation

127

110
111
112
113
114
115
116
117
118
119
120
121
122
126

gl.drawArrays(gl.TRIANGLES, 0, n);
}
// Last time when this function was called
var g_last = Date.now();
function animate(angle) {
// Calculate the elapsed time
var now = Date.now();
var elapsed = now - g_last; // milliseconds
g_last = now;
// Update the current rotation angle (adjusted by the elapsed time)
var newAngle = angle + (ANGLE_STEP * elapsed) / 1000.0;
return newAngle %= 360;
}

Line 7 in the vertex shader is just a multiplication of a matrix and the vertex coordinates
in the same way as RotatedTranslatedTriangle.js). u_ModelMatrix is a uniform variable,
and the rotation matrix is passed to the variable from a JavaScript program:
7

'

gl_Position =

u_ModelMatrix * a_Position;\n' +

The variable ANGLE_STEP at line 17 defines the rotation angles per second and is set to 45
degrees/second:
17

var ANGLE_STEP = 45.0;

The main() function starts from line 19, but because the code from lines 19 to 37, which
specifies the vertex coordinates, is the same as before, it is omitted.
The first of the three differences is that you specify the clear color once only: at line 44.
Line 47 then retrieves the storage location of u_ModelMatrix in the vertex shader. Because
this location never changes, it’s more efficient to do only this once:
47

var u_ModelMatrix = gl.getUniformLocation(gl.program, 'u_ModelMatrix');

The variable u_ModelMatrix is then used in the draw() function (line 99) that draws the
triangle.
The value of the variable currentAngle starts at 0 degrees and stores how many degrees
the triangle should be rotated from its original position each time it is drawn. As in the
simple rotation examples earlier, it calculates the rotation matrix needed for the transformation. The variable modelMatrix defined at line 56 is a Matrix4 object used to hold
the rotation matrix in draw(). This matrix could be created within draw(); however, that
would require a new Matrix4 object to be created each time draw() is called, which would
be inefficient. For this reason, the object is created at line 56 and then passed to draw() at
line 61.

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Lines 59 to 64 implement Mechanism 1 as the function tick, which is repeatedly called
to draw the triangle. Before you look at how the overall “tick” mechanism actually works,
let’s look at what happens each time it is called:
53
54
55
56
57
58
59
60
61
62
63
64

// Current rotation angle of a triangle
varcurrentAngle = 0.0;
// Matix4 object for model transformation
var modelMatrix = new Matrix4();
// Start to draw a triangle
var tick = function() {
currentAngle = animate(currentAngle); // Update the rotation angle
draw(gl, n, currentAngle, modelMatrix, u_ModelMatrix);
requestAnimationFrame(tick);// Request that the browser callstick
};
tick();

<- (2)

Within tick, the call to the function animate() at line 60 updates the current rotation
angle of the triangle, and then the call to draw() at line 61 draws a triangle using gl.
drawArrays().
draw() is passed the rotation matrix, which rotates the triangle to currentAngle degrees.
In turn, it passes the matrix to the u_ModelMatrix variable in the vertex shader before
calling gl.drawArrays() (lines 104 to 110). This code appears quite complex, so let’s

examine each part in turn.

Repeatedly Call the Drawing Function (tick())
As described earlier, to animate the triangle, you need to perform the following two steps
repeatedly: (1) update the current rotation angle of a triangle (currentAngle), and then (2)
call the drawing function with the angle to draw the triangle. Lines 59 to 64 implement
these processing steps.
In this sample program, these tasks are defined by the three operations of line 60, 61,
and 62. These operations are grouped in a single anonymous function using function(),
and the function is assigned to the variable tick (see Figure 4.8). You use an anonymous
function if you want to pass the local variables defined in main() (gl, n, currentAngle,
modelMatrix, and u_ModelMatrix) to draw() as arguments when draw() is called at line 61.
If you need a refresher on anonymous functions, refer to Chapter 2, “Your First Step with
WebGL,” where you used one to register an event handler.

Animation

129

Update the current rotation angle
( line 60 )

Draw a triangle using the angle
( line 61 )

Request that the browser calls this function
( tick() ) again ( line 62 )

Figure 4.8

The operations assigned to “tick”

You can use this basic approach for all types of animation. It is a key technique in 3D
graphics.
When you call requestAnimationFrame() at line 62, you are requesting the browser to call
the function specified as the first parameter at some future time when the three operations assigned to tick will be executed again. You’ll look at requestAnimationFrame() in a
moment. For now, let’s finish examining the operations executed in tick().

Draw a Triangle with the Specified Rotation Angle (draw())
The draw() function takes the following five parameters:

• gl: The context in which to draw the triangle
• n: The number of vertices
• currentAngle: The current rotation angle
• modelMatrix: A Matrix4 object to store the rotation matrix calculated using currentAngle

• u_ModelMatrix: The location of the uniform variable to which the modelMatrix is
passed
The actual function code is found in lines 99 to 111:
99
100
101
102
103
104
105
106
107

130

function draw(gl, n, currentAngle, modelMatrix,
// Set the rotation matrix
modelMatrix.setRotate(currentAngle, 0, 0, 1);

u_ModelMatrix) {

// Pass the rotation matrix to the vertex shader
gl.uniformMatrix4fv( u_ModelMatrix, false, modelMatrix.elements);
// Clear 
gl.clear(gl.COLOR_BUFFER_BIT);

CHAPTER 4

More Transformations and Basic Animation

108
109
110
111

// Draw the triangle
gl.drawArrays(gl.TRIANGLES, 0, n);
}

First, line 101 calculates the rotation matrix using the setRotate() method provided by
cuon-matrix.js, writing the resulting matrix to modelMatrix:
101 modelMatrix.setRotate(currentAngle, 0, 0, 1);

Next, line 104 passes the matrix to the vertex shader by using gl.uniformMatrix4fv():
104 gl.uniformMatrix4fv(u_ModelMatrix, false, modelMatrix.elements);

After that, line 107 clears the  and then calls gl.drawArrays() at line 110 to
execute the vertex shader to actually draw the triangle. Those steps are the same as used
before.
Now let’s return to the third operation, requestAnimationFrame(), which requests the
browser to call the function tick() at some future time.

Request to Be Called Again (requestAnimationFrame())
Traditionally, if you wanted to repeatedly execute specific tasks (functions) in JavaScript,
you used the method setInterval().

setInterval(func, delay)
Call the function specified by func multiple times with intervals specified by delay.
Parameters

Return value

func

Specifies the function to be called multiple times.

delay

Specifies the intervals (in milliseconds).

Timer id

However, because this JavaScript method was designed before browsers started to support
multiple tabs, it executes regardless of which tab is active. This can lead to performance
problems, so a new method, requestAnimationFrame(), was recently introduced. The
function scheduled using this method is only called when the tab in which it was defined
is active. Because requestAnimationFrame() is a new method and not yet standardized,
it is defined in the library supplied by Google, webgl-utils.js, which handles the differences among different browsers.

Animation

131

requestAnimationFrame(func)
Requests the function specified by func to be called on redraw (see Figure 4.9). This
request needs to be remade after each callback.
Parameters

func

Specifies the function to be called later. The function takes a
“time” parameter, indicating the timestamp of the callback.

Return value

Request id

Request the browser to call
tick() when the browser
redraws the screen.

JavaScript

…
requestAnimationFrame(tick ) ;
…

canvas
function tick() {
…
}

When the browser
redraws the screen,
the browser calls
tick().

Figure 4.9

The requestAnimationFrame () mechanism

By using this method, you avoid animation in inactive tabs and do not increase the load
on the browser. Note, you cannot specify an interval before the function is called; rather,
func (the first parameter) will be called when the browser wants the web page containing the element (the second parameter) to be painted. In addition, after calling the function, you need to request the callback again because the previous request is automatically
removed once it’s fulfilled. Line 62 makes that request again once tick is called and
makes it possible to call tick() repeatedly:
62

requestAnimationFrame(tick); // Request the browser to call tick

If you want to cancel the request to call the function, you need to use
cancelAnimationFrame().

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cancelAnimationFrame(requestID)
Cancel the function registered by requestAnimationFrame().
Parameter

requestID

Return value

None

Specifies the return value of requestAnimationFrame().

Update the Rotation Angle (animate())
Finally, let’s see how to update the current rotation angle. The program maintains the
current rotation angle of the triangle (that is, how many degrees the triangle has been
rotated from its original position) in the variable currentAngle (defined at line 54). It
calculates the next rotation angle based on this current value.
The update of currentAngle is carried out in the function animate(), which is called at
line 60. This function, defined at line 115, takes one parameter, angle, which represents
the current rotation angle and returns the new rotation angle:
60
61
113
114
115
116
117
118
119
120
121
122
123

currentAngle = animate(currentAngle);// Update the rotation angle
draw(gl, n, currentAngle, modelMatrix, u_ModelMatrix);
...
// Last time this function was called
var g_last = Date.getTime();
function animate(angle) {
// Calculate the elapsed time
var now = Date.getTime();
var elapsed = now - g_last;
g_last = now;
// Update the current rotation angle (adjusted by the elapsed time)
var newAngle = angle + (ANGLE_STEP * elapsed) / 1000.0;
return newAngle %= 360;
}

The process for updating the current rotation angle is slightly complicated. Let’s look at
the reason for that by using Figure 4.10.

Animation

133

i1

i0
t0

t1

i0 is not equal to i1
t2

t3

t4
time

Figure 4.10

The interval times between each tick() vary

Figure 4.10 illustrates the following:

•

tick() is called at t0. It calls draw() to draw the triangle and then reregisters tick().

•

tick() is called at t1. It calls draw() to draw the triangle and then reregisters tick().

•

tick() is called at t2. It calls draw() to draw the triangle and then reregisters tick().

The problem here is that the interval times between t0 and t1, t1 and t2, and t2 and t3
may be different because of the load on the browser at that time. That is, t1 – t0 could
be different from t2 – t1.
If the interval time is not constant, then simply adding a fixed amount of angle (degree/
second) to the current rotation angle each time tick() is called will result in an apparent
acceleration or deceleration of the rotation speed.
For this reason, the function animate() needs to be a little more sophisticated and must
determine the new rotation angle based on how long it has been since the function was
last called. To do that, you need to store the time that the function was last called into the
variable g_last and store the current time into the variable now. Then you can calculate
how long it has been since the function was last called by subtraction and store the result
in the variable elapsed (line 118). The amount of rotation is then calculated at line 121
using elapsed as follows.
121 varnewAngle = angle + (ANGLE_STEP * elapsed) / 1000.0;

The variables g_last and now contain the return value from the method now() of a Date
object whose units are a millisecond (1/1000 second). Therefore, if you want to rotate
the triangle by ANGLE_STEP (degree/second), you just need to multiply ANGLE_STEP by
elapsed/1000 to calculate the rotation angle. At line 121, you actually multiply ANGLE_
STEP by elapsed and then divide the result by 1000 because this is slightly more accurate,
but both have the same meaning.
Finally, line 122 ensures the value of newAngle is less than 360 (degrees) and returns the
result.

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If you now load RotatedTriangle.html into your browser, you can check that the triangle
rotates at a constant speed. We will reuse this approach for animation in the following
chapters, so it’s worthwhile making sure you have mastered the details.

Experimenting with the Sample Program
In this section, let’s create an animation that consists of multiple transformations.
RotatingTranslatedTriangle translates a triangle 0.35 units in the positive direction of
the x-axis first and then rotates the triangle by 45 degrees/second.
This is easy to achieve if you remember that multiple transformations can be realized by
multiplying each transformation matrix together (refer to Chapter 3).
To do this, you just need to insert the translation at line 102. Because the variable modelMatrix already contains the rotation matrix, you can use translate(), rather than
setTranslate(), to multiply modelMatrix by the translation matrix:
99
100
101
102
103
104

function draw(gl, n, currentAngle, modelMatrix, u_ModelMatrix) {
// Set a rotation matrix
modelMatrix.setRotate(currentAngle, 0, 0, 1);
modelMatrix.translate(0.35, 0, 0);
// Pass the rotation matrix to the vertex shader
gl.uniformMatrix4fv( u_ModelMatrix, false, modelMatrix.elements);

If you load the example, you will see the animation shown in Figure 4.11.

Figure 4.11

Multiple overlaid screenshots of RotatingTranslatedTriangle

Finally, for those of you wanting a little control, on the companion site for this book is
a sample program, named RotatingTriangle_withButtons, that allows dynamic control
of the rotation speed using buttons (see Figure 4.12). You can see the buttons below the
.

Animation

135

Figure 4.12

RotatingTriangle_withButtons

Summary
This chapter explored the process of transforming shapes using the transformation matrix
library, combining multiple basic transformations to create a complex transformation, and
animating shapes using the library. There are two key lessons in this chapter: (1) Complex
transformations can be realized by multiplying a series of basic transformation matrices;
(2) You can animate shapes by repeating the transformation and drawing steps.
Chapter 5, “Using Colors and Texture Images,” is the last chapter that covers basic techniques. It explores colors and textures. Once you master those, you will have enough
knowledge to create your own basic WebGL programs and will be ready to begin exploring
some of the more advanced capabilities of WebGL.

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Chapter 5

Using Colors and Texture Images

The previous chapters explained the key concepts underlying the foundations of
WebGL through the use of examples based on 2D shapes. This approach has given you
a good understanding of how to deal with single color geometric shapes in WebGL.
Building on these basics, you now delve a little further into WebGL by exploring the
following three subjects:

• Passing other data such as color information to the vertex shader
• The conversion from a shape to fragments that takes place between the vertex
shader and the fragment shader, which is known as the rasterization process

• Mapping images (or textures) onto the surfaces of a shape or object
This is the final chapter that focuses on the key functionalities of WebGL. After reading
this chapter, you will understand the techniques and mechanism for using colors
and textures in WebGL and will have mastered enough WebGL to allow you to create
sophisticated 3D scenes.

Passing Other Types of Information to Vertex
Shaders
In the previous sample programs, a single buffer object was created first, the vertex
coordinates were stored in it, and then it was passed to the vertex shader. However,
beside coordinates, vertices involved in 3D graphics often need other types of information such as color information or point size. For example, let us take a look at a
program you used in Chapter 3, “Drawing and Transforming Triangles,” which draws
three points: MultiPoint.js. In the shader, in addition to the vertex coordinates, you
provided the point size as extra information. However, the point size was a fixed value
and set in the shader rather than passed from outside:

3 var VSHADER_SOURCE =
4 'attribute vec4 a_Position;\n' +
5 'void main() {\n' +
6 ' gl_Position = a_Position;\n' +
7 ' gl_PointSize = 10.0;\n' +
8 '}\n'

Line 6 assigns the vertex coordinates to gl_Position, and line 7 assigns a fixed point
size of 10.0 to gl_PointSize. If you now wanted to modify the size of that point from
your JavaScript program, you would need a way to pass the point size with the vertex
coordinates.
Let’s look at an example, MultiAttributeSize, whose goal is to draw three points of
different sizes: 10.0, 20.0, and 30.0, respectively (see Figure 5.1).

Figure 5.1

MultiAttributeSize

In the previous chapter, you carried out the following steps to pass the vertex coordinates:
1. Create a buffer object.
2. Bind the buffer object to the target.
3. Write the coordinate data into the buffer object.
4. Assign the buffer object to the attribute variable.
5. Enable the assignment.
If you now wanted to pass several items of vertex information to the vertex shader
through buffer objects, you could just apply the same steps to all the items of information
associated with a vertex. Let’s look at a sample program that uses multiple buffers to do
just that.
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Sample Program (MultiAttributeSize.js)
MultiAttributeSize.js is shown in Listing 5.1. The fragment shader is basically the same
as in MultiPoint.js, so let’s omit it this time. The vertex shader is also similar, apart from

the fact that you add a new attribute variable that specifies the point size. The numbers 1
through 5 on the right of the listing note the five steps previously outlined.
Listing 5.1

MultiAttributeSize.js

1 // MultiAttributeSize.js
2 // Vertex shader program
3 var VSHADER_SOURCE =
4
'attribute vec4 a_Position;\n' +
5
'attribute float a_PointSize;\n' +
6
'void main() {\n' +
7
' gl_Position = a_Position;\n' +
8
' gl_PointSize = a_PointSize;\n' +
9
'}\n';
...
17 function main() {
...
34
// Set the vertex information
35
var n = initVertexBuffers(gl);
...
47
// Draw three points
48
gl.drawArrays(gl.POINTS, 0, n);
49 }
50
51 function initVertexBuffers(gl) {
52
var vertices = new Float32Array([
53
0.0, 0.5,
-0.5, -0.5,
0.5, -0.5
54
]);
55
var n = 3;
56
57
var sizes = new Float32Array([
58
10.0, 20.0, 30.0 // Point sizes
59 ]);
60
61
// Create a buffer object
62
var vertexBuffer = gl.createBuffer();
63
var sizeBuffer = gl.createBuffer();
...
69
// Write vertex coordinates to the buffer object and enable it
70
gl.bindBuffer(gl.ARRAY_BUFFER, vertexBuffer);
71
gl.bufferData(gl.ARRAY_BUFFER, vertices, gl.STATIC_DRAW);
72
var a_Position = gl.getAttribLocation(gl.program, 'a_Position');

<-(1)
<-(1')

<-(2)
<-(3)

Passing Other Types of Information to Vertex Shaders

139

77
78
79
80
81
82
83
88
89
94
95

...
gl.vertexAttribPointer(a_Position, 2, gl.FLOAT, false, 0, 0);
gl.enableVertexAttribArray(a_Position);
// Write point sizes to the buffer object and enable it
gl.bindBuffer(gl.ARRAY_BUFFER, sizeBuffer);
gl.bufferData(gl.ARRAY_BUFFER, sizes, gl.STATIC_DRAW);
var a_PointSize = gl.getAttribLocation(gl.program, 'a_PointSize');
...
gl.vertexAttribPointer(a_PointSize, 1, gl.FLOAT, false, 0, 0);
gl.enableVertexAttribArray(a_PointSize);
...
return n;
}

<-(4)
<-(5)

<-(2')
<-(3')

<-(4')
<-(5')

First of all, let us examine the vertex shader in Listing 5.1. As you can see, the attribute
variable a_PointSize, which receives the point size from the JavaScript program, has been
added. This variable, declared at line 5 as a float, is then assigned to gl_PointSize at line
8. No other changes are necessary for the vertex shader, but you will need a slight modification to the process in initVertexBuffers() so it can handle several buffer objects. Let
us take a more detailed look at it.

Create Multiple Buffer Objects
The function initVertexBuffers() starts at line 51, and the vertex coordinates are
defined from lines 52 to 54. The point sizes are then specified at line 57 using the array
sizes:
57
58
59

var sizes = new Float32Array([
10.0, 20.0, 30.0 // Point sizes
]);

A buffer object is created at line 62 for the vertex data, and at line 63 another buffer object
(sizeBuffer) is created for storing the array of “point sizes.”
From lines 70 to 78, the program binds the buffer object for the vertex coordinates, writes
the data, and finally assigns and enables the attribute variables associated with the buffer
object. These tasks are the same as those described in the previous sample programs.
Lines 80 to 89 are new additions for handling the different point sizes. However, the steps
are the same as for a vertex buffer. Bind the buffer object for the point sizes (sizeBuffer)
to the target (line 81), write the data (line 82), assign the buffer object to the attribute
variable a_PointSize (line 88), and enable it.
Once these steps in initVertexBuffers() are completed, the internal state of the WebGL
system looks like Figure 5.2. You can see that the two separate buffer objects are created
and then assigned to the two separate attribute variables.

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Using Colors and Texture Images

function m ain() {
var gl=getW ebGL…
…
initShaders( … ) ;
…
}

0.0, 0.5
-0.5, -0.5
0.5, -0.5

Vertex Shader

sizeBuffer
B u ffe r O b je c t

10.0,
20.0,
30.0

Figure 5.2

a ttr ib u te a _ P o in tS Iz e

g l_ P o s itio n =
a _ P o s itio n ;
g l_ P o in tS iz e =
a _ P o in tS iz e ;

WebGL System

Fragment
Shader
g l_ F r a g C o lo r =
v e c 4 (… ) ;

gl_FragColor

B u ffe r O b je c t

a ttr ib u te a _ P o s itio n

gl_Position

vertexBuffer

gl_PointSize

Ja va S cript

C o lo r B u ffe r

Using two buffer objects to pass data to a vertex shader

In this situation, when gl.drawArrays() at line 48 is executed, all the data stored inside
the buffer objects is sequentially passed to each attribute variable in the order it was stored
inside the buffer objects. By assigning this data to gl_Position at line 7 and gl_PointSize
at line 8, respectively (the vertex shader’s program in Figure 5.2), you are now able to
draw different size objects located at different positions.
By creating a buffer object for each type of data in this way and then allocating it to the
attribute variables, you can pass several pieces of information about each vertex to the
vertex shader. Other types of information that can be passed include color, texture coordinates (described in this chapter), and normals (see Chapter 7), as well as point size.

The gl.vertexAttribPointer() Stride and Offset Parameters
Although multiple buffer objects are a great way to handle small amounts of data, in a
complicated 3D object with many thousands of vertices, you can imagine that managing
all the associated vertex data is an extremely difficult task. For example, imagine needing
to manually check each of these arrays when the total count of MultiAttributeSize.js’s
vertices and sizes reaches 1000.1 However, WebGL allows the vertex coordinates and the
size to be bundled into a single component and provides mechanisms to access the different data types. For example, you can group the vertex and size data in the following way
(refer to Listing 5.2), often referred to as interleaving.
Listing 5.2

An Array Containing Multiple Items of Vertex Information

var verticesSizes = new Float32Array([
// Vertex coordinates and size of a point
0.0, 0.5, 10.0,
// The 1st point
-0.5, -0.5, 20.0, // The 2nd point
0.5, -0.5, 30.0
// The 3rd point
]);

1

In practice, because modeling tools that create 3D models actually generate this data, there is no
necessity to either manually input them or visually check their consistency. The use of modeling
tools and the data they generate will be discussed in Chapter 10.

Passing Other Types of Information to Vertex Shaders

141

As just described, once you have stored several types of information pertaining to the
vertex in a single buffer object, you need a mechanism to access these different data
elements. You can use the fifth (stride) and sixth (offset) arguments of gl.vertexAttribPointer() to do this, as shown in the example that follows.

Sample Program (MultiAttributeSize_Interleaved.js)
Let’s construct a sample program, MultiAttributeSize_Interleaved, which passes multiple data to the vertex shader, just like MultiAttributeSize.js (refer to Listing 5.1), except
that it bundles the data into a single array or buffer. Listing 5.3 shows the program in
which the vertex shader and the fragment shader are the same as in MultiAttributeSize.
js.
Listing 5.3

MultiAttributeSize_Interleaved.js

1 // MultiAttributeSize_Interleaved.js
2 // Vertex shader program
3 var VSHADER_SOURCE =
4
'attribute vec4 a_Position;\n' +
5
'attribute float a_PointSize;\n' +
6
'void main() {\n' +
7
' gl_Position = a_Position;\n' +
8
' gl_PointSize = a_PointSize;\n' +
9
'}\n';
...
17 function main() {
...
34
// Set vertex coordinates and point sizes
35
var n = initVertexBuffers(gl);
...
48
gl.drawArrays(gl.POINTS, 0, n);
49 }
50
51 function initVertexBuffers(gl) {
52
var verticesSizes = new Float32Array([
53
// Vertex coordinates and size of a point
54
0.0, 0.5, 10.0, // The 1st vertex
55
-0.5, -0.5, 20.0, // The 2nd vertex
56
0.5, -0.5, 30.0
// The 3rd vertex
57
]);
58
var n = 3;
59
60
// Create a buffer object
61
var vertexSizeBuffer = gl.createBuffer();
...

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67
68
69
70
71
72
73

// Write vertex coords and point sizes to the buffer and enable it
gl.bindBuffer(gl.ARRAY_BUFFER, vertexSizeBuffer);
gl.bufferData(gl.ARRAY_BUFFER, verticesSizes, gl.STATIC_DRAW);

78
79
80
81
82

var FSIZE = verticesSizes.BYTES_PER_ELEMENT;
// Get the storage location of a_Position, allocate buffer, & enable
var a_Position = gl.getAttribLocation(gl.program, 'a_Position');
...
gl.vertexAttribPointer(a_Position, 2, gl.FLOAT, false, FSIZE * 3, 0);
gl.enableVertexAttribArray(a_Position); // Enable allocation

87
88

// Get the storage location of a_PointSize, allocate buffer, & enable
var a_PointSize = gl.getAttribLocation(gl.program, 'a_PointSize');
...
gl.vertexAttribPointer(a_PointSize, 1, gl.FLOAT, false, FSIZE * 3, FSIZE * 2);
gl.enableVertexAttribArray(a_PointSize); // Enable buffer allocation
...
return n;

93
94 }

The processing flow of the main() function in JavaScript is the same as
MultiAttributeSize.js, and only the initVertexBuffers() process is modified this time,
so let’s take a look at its content.
First, a typed array is defined at lines 52 to 57, as previously described in Listing 5.2.
Following the usual processing steps, from line 61 to 69, a buffer object is created (line
61), the object is bound (line 68), and the data is written to the object (line 69). Next,
at line 71, the size (number of bytes) of the element in the verticeSizes array is stored
in the variable FSIZE, which will be needed later on. The size (number of bytes) of each
element of a typed array can be obtained through the property BYTES_PER_ELEMENT.
From line 73 onward, you assign the buffer object to the attribute variable. Retrieving the
storage location of the attribute variable a_Position at line 73 is similar to the previous
example, but the usage of the arguments of gl.vertexAttribPointer() at line 78 is different because the buffer now holds two types of data: vertex and point size.
You’ve already looked at the specification of gl.vertexAttribPointer() in Chapter 3, but
let’s take another look and focus on two parameters: stride and offset.

Passing Other Types of Information to Vertex Shaders

143

gl.vertexAttribPointer(location, size, type, normalized, stride,
offset)
Assign the buffer object bound to gl.ARRAY_BUFFER to the attribute variable specified by
location. The type and format of the data written in the buffer is also specified.
Parameters

location

Specifies the storage location of the attribute variable.

size

Specifies the number of components per vertex in the buffer
object (valid values are 1 to 4).

type

Specifies the data format (in this case, gl.FLOAT)

normalized

true or false. Used to indicate whether non-float data should

be normalized to [0, 1] or [–1, 1].
stride

Specifies the stride length (in bytes) to get vertex data; that is,
the number of bytes between each vertex element

offset

Specifies the offset (in bytes) in a buffer object to indicate where
the vertex data is stored from. If the data is stored from the
beginning. then offset is 0.

The stride specifies the number of bytes used by a group of related vertex data (in this
example, vertex coordinates and point size) inside the buffer object.
In previous examples, where you had only one type of information in the buffer—
vertices—you set the stride to 0. However, in this example, both vertices and point sizes
are laid out in the buffer, as shown in Figure 5.3.
FSIZE = verticesSizes.BYTES_PER_ELEMENT
The offset of
a_Position(0)

0 .0
0 .5

The offset of
a_PointSize
(FSIZE* 2)

stride(FSIZE * 3)

1 0 .0
-0 .5
-0 .5
2 0 .0

stride(FSIZE * 3)
NOTE : This figure is drawn assuming FSIZE=4

0 .5

Figure 5.3

Stride and offset

As illustrated in Figure 5.3, there are three components inside each group of vertex data
(two coordinates, one size), so you need to set the stride equal to three times the size of
each component in the group (that is, three times FSIZE [the number of bytes per element
of the Floats32Array]).

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The offset parameter indicates the distance to the first element that is being used for this
call. Because you are using the vertex coordinates that are positioned at the head of the
verticesSizes array, the offset is 0. So, at line 78, you specify them as the fifth (stride) and
sixth (offset) arguments of gl.vertexAttribPointer():
78
79

gl.vertexAttribPointer(a_Position, 2, gl.FLOAT, false, FSIZE * 3, 0);
gl.enableVertexAttribArray(a_Position); // Enable allocation

Finally, once the specification of the vertex coordinates has been set up, the assignment to
a_Position is enabled at line 79.
Next, from line 82, you need to do the same for the point size data, so assign a buffer
object to a_PointSize. However, in this case, you are using the same buffer that you used
for vertex data, but you want different data from the buffer. You can make use of the sixth
argument offset to achieve this by setting the offset to the location at which the data (in
this case the point size) to be passed to a_PointSize is positioned in the buffer. The first
two elements of the array are vertex coordinates, so the offset will accordingly be set to
FSIZE * 2 (refer to Figure 5.3). Line 87 shows both stride and offset set up correctly:
87
88

gl.vertexAttribPointer(a_PointSize, 1, gl.FLOAT, false, FSIZE * 3, FSIZE * 2);
gl.enableVertexAttribArray(a_PointSize); // Enable buffer allocation

The assignment to a_PointSize is enabled at line 88, and the only remaining task to
perform is the draw operation using gl.drawArrays().
Each time a vertex shader is invoked, WebGL will extract data from the buffer object using
the values specified in stride and offset and subsequently pass them to the attribute variables to be used for drawing (see Figure 5.4).
a ttrib u te a _ P o s itio n

Ja va S cript

Figure 5.4

Vertex Shader

- 0 . 5, - 0 . 5 , 2 0 . 0 ,
0 . 5 , - 0 . 5 , 3 0 .0 ,

g l_ P o s itio n =
a _ P o s it io n ;
g l_ P o in tS iz e =
a _ P o in tS iz e ;

Fragment
Shader
g l_ F ra g C o lo r =
v e c 4 ();

gl_FragColor

B u ffe r O b je c t

0.0, 0.5, 10.0,

gl_PointSize

function main() {
var gl=getWebGL…
…
initShaders(…);
…
}

gl_Position

a ttr ib u te a _ P o in tS iz e

Color Buffer

Internal behavior when stride and offset are used

Passing Other Types of Information to Vertex Shaders

145

Modifying the Color (Varying Variable)
Now that you have seen how to pass several pieces of information to the vertex shader,
let’s use the technique to modify the color of each point. You can achieve this using the
procedure explained previously, substituting color information for point size in the buffer.
After storing the vertex coordinates and the color in the buffer object, you will assign the
color to the attribute variable, which handles the color.
Let’s construct a sample program, MultiAttributeColor, that draws red, blue, and green
points. A screenshot is shown in Figure 5.5. (Because this book is black and white, it might
be difficult to appreciate the difference between the colors, so load and run the code in
your browser.)

Figure 5.5

MultiAttributeColor

As you may remember from Chapter 2, “Your First Step with WebGL,” the fragment
shader actually handles attributes like color. Up until this point, you’ve set up color statically in the fragment shader code and not touched it again. However, although you have
learned how to pass the point color information to the vertex shader through the attribute variable, the use of the gl_FragColor variable, which sets the color information, is
restricted to the fragment shader. (Refer to the section “Fragment Shader” in Chapter 2.)
Therefore, you need to find a way to communicate to the fragment shader the color information previously passed to the vertex shader (Figure 5.6).

146

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Using Colors and Texture Images

a ttrib u te a _ P o s itio n ;
a ttrib u te a _ C o lo r ;

Vertex Shader

B u ffe r O b je c t

0.0, 0.5, 1.0, 0.0, 0.0,
-0.5, -0.5, 0.0,1.0, 0.0,
0.5, -0.5, 0.0, 0.0, 1.0

g l_ P os i t i o n =
a_Posit ion;

gl_Position

function main() {
var gl=getWebGL…
…
initShaders(…);
…
}

WebG L System
F ra g m e n t
Shader

gl_FragColor

Ja va S cript

g l_ F r a g C o lo r

Color Buffer
H ow can w e
p a s s d a ta ?

Figure 5.6

Passing data from a vertex shader to a fragment shader

In ColoredPoints (Chapter 2), a uniform variable was used to pass the color information to the fragment shader; however, because it is a “uniform” variable (not varying),
it cannot be used to pass different colors for each vertex. This is where a new method to
pass data to the fragment shader through the varying variable is needed and relies on a
mechanism that sends data from the vertex shader to the fragment shader: by using the
varying variable. Let’s look at a concrete sample program.

Sample Program (MultiAttributeColor.js)
Listing 5.4 shows the program, which looks similar to the program introduced in the
previous section, MultiAttributeSize_Stride.js, but the part related to the vertex and
fragment shaders is actually slightly different.
Listing 5.4
1
2
3
4
5
6
7
8
9
10
11
12
13
14

MultiAttributeColor.js

// MultiAttributeColor.js
// Vertex shader program
var VSHADER_SOURCE =
'attribute vec4 a_Position;\n' +
'attribute vec4 a_Color;\n' +
'varying vec4 v_Color;\n' + // varying variable
'void main() {\n' +
' gl_Position = a_Position;\n' +
' gl_PointSize = 10.0;\n' +
' v_Color = a_Color;\n' + // Pass the data to the fragment shader
'}\n';

// Fragment shader program
var FSHADER_SOURCE =
...
18 'varying vec4 v_Color;\n' +
19
'void main() {\n' +

Passing Other Types of Information to Vertex Shaders

147

20
' gl_FragColor = v_Color;\n' + // Receive the data from the vertex shader
21
'}\n';
22
23 function main() {
...
40
// Set vertex coordinates and color
41
var n = initVertexBuffers(gl);
...
54
gl.drawArrays(gl.POINTS, 0, n);
55 }
56
57 function initVertexBuffers(gl) {
58
var verticesColors = new Float32Array([
59
// Vertex coordinates and color
60
0.0, 0.5, 1.0, 0.0, 0.0,
61
-0.5, -0.5, 0.0, 1.0, 0.0,
62
0.5, -0.5, 0.0, 0.0, 1.0,
63
]);
64
var n = 3; // The number of vertices
65
66
// Create a buffer object
67
var vertexColorBuffer = gl.createBuffer();
...
73
// Write the vertex coordinates and colors to the buffer object
74
gl.bindBuffer(gl.ARRAY_BUFFER, vertexColorBuffer);
75
gl.bufferData(gl.ARRAY_BUFFER, verticesColors, gl.STATIC_DRAW);
76
77
var FSIZE = verticesColors.BYTES_PER_ELEMENT;
78
// Get the storage location of a_Position, allocate buffer, & enable
79
var a_Position = gl.getAttribLocation(gl.program, 'a_Position');
...
84
gl.vertexAttribPointer(a_Position, 2, gl.FLOAT, false, FSIZE * 5, 0);
85
gl.enableVertexAttribArray(a_Position); // Enable buffer assignment
86
87
// Get the storage location of a_Color, assign buffer, and enable
88
var a_Color = gl.getAttribLocation(gl.program, 'a_Color');
...
93 gl.vertexAttribPointer(a_Color, 3, gl.FLOAT, false, FSIZE*5, FSIZE*2);
94 gl.enableVertexAttribArray(a_Color); // Enable buffer allocation
...
96
return n;
97 }

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At line 5 of the vertex shader, an attribute variable a_Color is declared in order to receive
the color data. Next, at line 6, a new varying variable v_Color is declared that will be used
to pass its value to the fragment shader. Please note that you can only use float types (and
related types vec2, vec3, vec4, mat2, mat3, and mat4) for varying variables:
5
6

'attribute vec4 a_Color;\n' +
'varying vec4 v_Color;\n' +

At line 10, the value of a_Color is assigned to the variable v_Color declared at line 6:
10

'

v_Color = a_Color;\n' +

So how can the fragment shader receive the assigned data? The answer is straightforward.
All that is required is declaring a variable in the fragment shader with the same name and
types as that in the vertex shader:
18

'varying vec4 v_Color;\n' +

In WebGL, when varying variables declared inside the fragment shader have identical
names and types to the ones declared in the vertex shader, the assigned values in the
vertex shader are automatically passed to the fragment shader (see Figure 5.7).
a ttrib u te a _ P o s itio n ;
a ttrib u te a _ C o lo r ;

B u ffe r O b je c t

0.0, 0.5, 1.0, 0.0, 0.0,
-0.5, -0.5, 0.0,1.0, 0.0,
0.5, -0.5, 0.0, 0.0, 1.0

Vertex Shader
g l_ P os i t i o n =
a_Posit ion;
v_Color = a_Color;

gl_Position

function main() {
var gl=getWebGL…
…
initShaders(…);
…
}

WebG L System
F ra g m e n t
Shader
g l_ F r a g C o lo r =
v_Color;

gl_FragColor

Ja va S cript

Color Buffer
v a ry in g v _ C o lor

Figure 5.7

v a ry in g v _C o lo r

The behavior of a varying variable

So, the fragment shader can receive the values assigned to the vertex shader at
line 10 simply by assigning the varying variable v_Color to gl_FragColor at line 20.
As gl_FragColor sets the fragment color, the color of each point will be modified:
20

'

gl_FragColor = v_Color;\n' +

The remaining code is similar to MultiAttributeSize.js. The only differences are that
the name of the typed array for vertex information defined at line 58 is modified to
verticesColors, and the color information such as (1.0, 0.0, 0.0) is added to the data
definition at line 60.

Passing Other Types of Information to Vertex Shaders

149

As previously explained in Chapter 2, the color information is specified using the 0.0–1.0
range for each component of the RGBA model. Just like MultiAttributeSize_Stride.
js, you store several different types of data within a single array. The fifth (stride) and
sixth (offset) arguments of gl.vertexAttribPointer() are modified at lines 84 and 93,
respectively, based on the content of the verticesColors array which, because you have
introduced some color information in addition to the vertex coordinates, means the stride
changes to FSIZE * 5.
Finally, the draw command at line 54 results in red, blue, and green points being
displayed in the browser.

Experimenting with the Sample Program
Let’s modify the first argument of gl.drawArrays() at line 54 to gl.TRIANGLES and see
what happens upon execution. Alternatively, you can load the ColoredTriangle sample
program from the book’s website:
54

gl.drawArrays(gl.TRIANGLES, 0, n);

The execution output is shown in Figure 5.8. It might be difficult to grasp the difference
when seen in black and white, but on your screen, notice that a nice smooth-shaded
triangle with red, green, and blue corners is drawn.

Figure 5.8

150

ColoredTriangle

CHAPTER 5

Using Colors and Texture Images

This significant change from three colored points to a smoothly shaded triangle occurred
just by changing one parameter value. Let’s look at how that came about.

Color Triangle (ColoredTriangle.js)
You already explored the subject of coloring triangles using a single color in Chapter 3.
This section explains how to specify a different color for each of the triangle’s vertices and
the process within WebGL that results in a smooth color transition between the different
vertices.
To fully comprehend the phenomenon, you need to understand in detail the process
carried out between the vertex and the fragment shaders, as well as the functionality of
the varying variable.

Geometric Shape Assembly and Rasterization
Let’s start the explanation using the example program, HelloTriangle.js, introduced in
Chapter 3, which simply draws a red triangle. The relevant code snippet necessary for the
explanation is shown in Listing 5.5.
Listing 5.5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

HelloTriangle.js (Code Snippet)

// HelloTriangle.js
// Vertex shader program
var VSHADER_SOURCE =
'attribute vec4 a_Position;\n' +
'void main() {\n' +
' gl_Position = a_Position;\n' +
'}\n';
// Fragment shader program
var FSHADER_SOURCE =
'void main() {\n' +
' gl_FragColor = vec4(1.0, 0.0, 0.0, 1.0);\n' +
'}\n';

function main() {
...
32
// Set vertex coordinates
33
var n = initVertexBuffers(gl);
...
45
// Draw a triangle
46
gl.drawArrays(gl.TRIANGLES, 0, n);
47 }
48
49 function initVertexBuffers(gl) {

Color Triangle (ColoredTriangle.js)

151

50
51
52
53

var vertices = new Float32Array([
0.0, 0.5,
-0.5, -0.5,
0.5, -0.5
]);
var n = 3; // The number of vertices
...
gl.vertexAttribPointer(a_Position, 2, gl.FLOAT, false, 0, 0);
...
return n;

74
81
82 }

In this program, after writing the vertex coordinates (lines 50 to 52) into the buffer object
in the function initVertexBuffers(), the buffer object is assigned to the attribute variable
a_Position at line 74. Following that, when gl.drawArrays() invokes the vertex shader at
line 46, the three vertex coordinates inside the buffer object are passed to a_Position at
line 4 and assigned to gl_Position at line 6, thus making them available to the fragment
shader. In the fragment shader, the RGBA value (1.0, 0.0, 0.0, 1.0) associated with the red
color is assigned to gl_FragColor, so a red triangle is displayed.
Up until now, you haven’t actually explored how this works, so let’s examine how exactly
a fragment shader performs per-fragment operations when you only give it the triangle’s
three vertex coordinates in gl_Position.
In Figure 5.9, you can see the problem. The program gives three vertices, but who identifies that the vertex coordinates assigned to gl_Position are the vertices of a triangle? In
addition, to make the triangle look like it is filled with a single color, who decides which
fragments have to be colored? Finally, who is responsible for invoking the fragment
shader and how it handles processing for each of the fragments?
F ra g m e n t
Shader
g l_ F r a g C o lo r =
v_C olor;

Figure 5.9 Vertex coordinate, identification of a triangle from the vertex coordinates,
rasterization, and execution of a fragment shader
Up until now, we have glossed over these details, but there are actually two processes
taking place between the vertex and the fragment shaders, which are shown in Figure
5.10.

• The geometric shape assembly process: In this stage, the geometric shape is assembled from the specified vertex coordinates. The first argument of gl.drawArray()
specifies which type of shape should be assembled.

• The rasterization process: In this stage, the geometric shape assembled in the
geometric assembly process is converted into fragments.

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v ar gl=getWebGL…
gl.drawArrays
( gl.TRIANGLES, 0, n);

B u ffe r
O b je c t

V e r te x S h a d e r

0.0, 0.5
-0.5, -0.5
0.5, -0.5

g l _ P os i t i o n =
a_ P o si t i o n ;

Shape
Assembly

Rasteriza
-tion

F ra g m e n t
Shader
gl_ F rag C olor =
vec4(...);

gl_FragColor

function main() {

gl_Position

a ttrib u te a _ P o s itio n

Ja va S cript

WebGL System

Figure 5.10

Assembly and rasterization between a vertex shader and a fragment shader

As you will have realized from Figure 5.10, gl_Position actually acts as the input to the
geometric shape assembly stage. Note that the geometric shape assembly process is also
called the primitive assembly process because the basic shapes previously shown in
Chapter 2 are also called primitives.
Figure 5.11 shows the processes between the vertex and fragment shaders, which are actually performed in assembly and rasterization for HelloTriangle.js.
From Listing 5.5, the third argument n of gl.drawArrays() (line 46) is set to 3, meaning
that the vertex shader is actually invoked three times.
Step 1. The vertex shader is invoked, and then the first coordinate (0.0, 0.5) inside the
buffer object is passed to the attribute variable a_Position. Once this is assigned
to gl_Position, this coordinate is communicated to the geometric shape assembly stage and held there. As you will remember, because only the x and y coordinates are passed to a_Position, the z and w values are supplied, so actually (0.0,
0.5, 0.0,1.0) is held.
Step 2. The vertex shader is once again invoked, and in a similar way the second coordinate (–0.5, –0.5) is passed to the geometric shape assembly stage and held there.
Step 3. The vertex shader is invoked a third time, passing the third coordinate (0.5, –0.5)
to the geometric shape assembly stage and holding it there.
Now the vertex shader processing is complete, and the three coordinates are
readily available for the geometric shape assembly stage.
Step 4. The geometric shape assembly processing starts. Using the three vertices
passed and the information (gl.TRIANGLES) contained in the first argument of
gl.drawArrays(), this stage decides how primitives should be assembled. In this
case, a triangle is assembled using the three vertices.
Step 5. Because what is displayed on the screen is a triangle consisting of fragments
(pixels), the geometric shape is converted to fragments. This process is called
rasterization. Here, the fragments that make up the triangle will be generated.
You can see the example of the generated fragments in the box of the rasterization stage in Figure 5.11.

Color Triangle (ColoredTriangle.js)

153

a ttrib u te a _ P os itio n

g l_Position =
a_Position;

gl_FragColor =
vec4(...);

a ttrib u te a _ P os itio n

Buffer
O b je c t

function main() {

Vertex Shader

var gl=getWebGL…
gl.drawArrays
(gl.TRIANGLES, 0, n);

0.0, 0.5
-0.5, -0.5
0.5, -0.5

Step(2)

g l_Position =
a_Position;

gl_Position

JavaScript

Shape
A ssembly

Rasteriza
-tion

Buffer
O b je c t

Vertex Shader

var gl=getWebGL…
gl.drawArrays
(gl.TRIANGLES, 0, n);

0.0, 0.5
-0.5, -0.5
0.5, -0.5

g l_Position =
a_Position;

Step(3)
gl_Position

function main() {

Shape
Assembly

Rasteriza
-tion

function main() {
var gl=getWebGL…
gl.drawArrays
(gl.TRIANGLE S, 0, n);

0.0, 0.5
-0.5, -0.5
0.5, -0.5

Step(4)

Vertex Shader

g l_Position =
a_Position;

gl_Position

Buffer
O b je c t

Shape
Assembly

Rasteriza
-tion

a ttrib u te a _ P os itio n

Vertex Shader

Step(5)

var gl=getWebGL…
gl.drawArrays
(gl.TRIANGLES, 0, n);

Figure 5.11

0.0, 0.5
-0.5, -0.5
0.5, -0.5

g l_Position =
a_Position;

gl_Position

Buffer
O b je c t

F ra g m e n t
Shader
g l_ Frag C olor =
vec4(...);

JavaScript
function main() {

F ra g m e n t
Shader
g l_ Frag C olor =
vec4(...);

a ttrib u te a _ P os itio n

JavaScript

F ra g m e n t
Shader
g l_ Frag C olor =
vec4(...);

a ttrib u te a _ P os itio n

JavaScript

F ra g m e n t
Shader

gl_FragColor

0.0, 0.5
-0.5, -0.5
0.5, -0.5

Rasteriza
-tion

gl_FragColor

gl.drawArrays
(gl.TRIANGLES, 0, n);

Shape
A ssembly

gl_FragColor

Ve rtex Shader

gl_FragColor

Buffer
O b je c t

var gl=getWebGL…

Shape
Assembly

Rasteriza
-tion

F ra g m e n t
Shader
g l_ Frag C olor =
vec4(...);

gl_FragColor

function main() {

Step(1)
gl_Position

JavaScript

The processing flow of geometric shape assembly and rasterization

In this figure, although we show only 10 fragments, the actual number of fragments is
determined according to the area where the triangle is finally displayed on the screen.
If you specify a different geometric shape in the first argument of gl.drawArrays(), the
geometric shape assembled in Step 4 is modified accordingly, as are the number of fragments and their position in Step 5. For example, if you specify gl.LINES, a line will be
assembled out of the first two coordinates, and the remaining one will be discarded. If you

154

CHAPTER 5

Using Colors and Texture Images

set it to gl.LINE_LOOP, a connected group of line segments will be generated, and only a
transparent (no fill color) triangle will be drawn.

Fragment Shader Invocations
Once the rasterization stage is completed, the fragment shader is invoked to process each
of the generated fragments. So in this example, the fragment shader is invoked 10 times,
as illustrated in Figure 5.12. To avoid cluttering the figure, we skip the intermediate steps.
All of the fragments are fed one by one to the fragment shader, and for each fragment,
the fragment shader sets the color and writes its output to the color buffer. When the
last fragment shader process is completed at Step 15, the final output is displayed in the
browser.

0.0, 0.5
-0.5, -0.5
0.5, -0.5

gl_Position =
a_Position;

gl_Position

Vertex
Shader

Buffer
Object

Shape
Assembly

Rasteriza
-tion

Fragment
Shader
gl_FragColor =
vec4(...);

gl_FragColor

Step(6)

attribute a_Position

Color Buffer

0.0, 0.5
-0.5, -0.5
0.5, -0.5

gl_Position =
a_Position;

gl_Position

Vertex
Shader

Buffer
Object

Shape
Assembly

Rasteriza
-tion

Fragment
Shader
gl_FragColor =
vec4(...);

gl_FragColor

Step(7)

attribute a_Position

Color Buffer

…

gl_Position =
a_Position;

Shape
Assembly

Rasteriza
-tion

Fragment
Shader
gl_FragColor =
vec4(...);

gl_FragColor

0.0, 0.5
-0.5, -0.5
0.5, -0.5

gl_Position

Vertex
Shader

Buffer
Object

Browser

Step(15)

attribute a_Position

Color Buffer

Figure 5.12

Fragment shader invocations

The following fragment shader in HelloTriangle.js colors each fragment in red. As a
result, a red filled triangle is written to the color buffer and displayed in the browser.
9 // Fragment shader program
10 var FSHADER_SOURCE =
11
'void main() {\n' +
12
' gl_FragColor = vec4(1.0, 0.0, 0.0, 1.0);\n' +
13
'}\n';

Color Triangle (ColoredTriangle.js)

155

Experimenting with the Sample Program
As an experiment, let’s confirm that the fragment shader is called for each fragment by
trying to set the color of each fragment based on its location. Each fragment generated by
the rasterization stage has its coordinates passed to the fragment shader upon invocation.
These coordinates can be accessed through the built-in variables provided inside the fragment shader (Table 5.1).
Table 5.1

Built-In Variables in a Fragment Shade (Input)

Type and Variable Name Description
vec4 gl_FragCoord

The first and second component are the coordinates of the fragment in the  coordinate system (window coordinate
system)

To check that the fragment shader is actually executed for each fragment, you can modify
line 12 in the program, as follows:
1 // HelloTriangle_FragCoord.js
...
9 // Fragment shader program
10 var FSHADER_SOURCE =
11 'precision mediump float;\n' +
12 'uniform float u_Width;\n' +
13 'uniform float u_Height;\n' +
14 'void main() {\n' +
15 ' gl_FragColor = vec4(gl_FragCoord.x/u_Width, 0.0, gl_FragCoord.y/u_Height,
➥1.0);\n' +
16 '}\n';

As you can see, the color components of each fragment, red and blue, are calculated based
on the fragment’s coordinates on the canvas. Note that the canvas’s y-axis is the inverse
direction to the WebGL coordinate system, and because it’s in WebGL, the color value
is expressed in the 0.0 to 1.0 range, you can divide the coordinates by the size of the
 element (that is, 400 pixels) to get the appropriate color value. As you can see,
the width and height are passed into the shader using the uniform variables u_Width and
u_Height and determined from gl.drawingBufferWidth and gl.drawingBufferHeight.
You can see the execution result in Figure 5.13, which is a triangle whose fragments are
colored as a function of their position. Running this sample program HelloTriangle_
FragCoord, you will see the transition from the left top to the right bottom.

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(0.0, 0.0)

x

Drawing area of

canvas

(400, 400)

y

Figure 5.13 Modifying the color per fragment (the figure on the right side shows the 
coordinate system)
Because you modify the color of each fragment with respect to its coordinates, you will
notice that the color progressively changes according to the coordinates. Again, if this is
not clear from the black-and-white image in Figure 5.13, run the example from the book’s
website.

Functionality of Varying Variables and the Interpolation Process
At this stage, you have a better understanding of the process flow taking place between
the vertex and the fragment shaders (that is, the geometric shape assembly process and
the subsequent rasterization process), as well as the invocation of the fragment shader for
each of the generated fragments.
Returning to Figure 5.8, the first ColoredTriangle, let’s use what you’ve learned to understand better why you get such a nicely shaded triangle when you specify a different color
for each the triangle’s vertex. Previously, you saw that the value assigned to the varying
variable inside the vertex shader is passed as a varying variable with identical attributes
(same name, same type) to the fragment shader (see Figure 5.14). However, to be more
precise, the value assigned to the varying variable in the vertex shader is interpolated at
the rasterization stage. Consequently, the value passed to the fragment shader actually
differs for each fragment based on that interpolation (see Figure 5.15). This is the reason
the varying variable has the name “varying.”

Color Triangle (ColoredTriangle.js)

157

a ttrib u te a _ P o s itio n ;
a ttrib u te a _ C o lo r ;

Vertex Shader

B u f fe r O b je c t

0.0, 0.5, 1.0, 0.0, 0.0,
-0.5, -0.5, 0.0,1.0, 0.0,
0.5, -0.5, 0.0, 0.0, 1.0

g l_ P os i t i o n =
a_Posit ion;
v_Color = a_Color;

gl_Position

function main() {
var gl=getWebGL…
…
initShaders(…);
…
}

WebG L System
F ra g m e n t
Shader
g l_ F r a g C o lo r =
v_Color;

gl_FragColor

Ja va S cript

Color Buffer
v a ry in g v _ C o lor

Figure 5.14

v a ry in g v _C o lo r

The behavior of a varying variable (reprint of Figure 5.7)

attribute a_Position;

B u f fe r O b je c t

S hape
A s s e m b ly

0.0, 0.5, 1.0, 0.0, 0.0,
-0.5, -0.5, 0.0,1.0, 0.0,
0.5, -0.5, 0.0, 0.0, 1.0

gl_Position =
a_Position;
v_Color = a_Color;

R a s t e r iz a
t io n

gl_Position

V e r te x S h a d e r

F ra g m e n t
S hader

gl_FragColor =
v_Color;

gl_FragColor

WebGL System

attribute a_Color;

Color Buffer
Interpola
tion

varying v_Color;

Figure 5.15

varying v_Color;

Interpolation of a varying variable

More specifically, in ColoredTriangle, because we only assign a different value to the
varying variable for each of the three vertices, each fragment located between vertices
must have its own color interpolated by the WebGL system.
For example, let’s consider the case in which the two end points of a line are specified
with different colors. One of the vertices is red (1.0, 0.0 0.0), whereas the other one is blue
(0.0, 0.0, 1.0). After the colors (red and blue) are assigned to the vertex shader’s v_Color,
the RGB values for each of the fragments located between those two vertices are calculated
and passed to the fragment shader’s v_Color (see Figure 5.16).
Red
(1.0, 0.0, 0.0)

B lu e
(0.0, 0.0, 1.0)

Rasterization

v_Color in a
vertex shader

Figure 5.16

158

Red (1.0, 0.0, 0.0)
Interpolated color (0.75, 0.0, 0.25)
Interpolated color (0.50, 0.0, 0.50)
Interpolated color (0.25, 0.0, 0.75)
Blue (0.0, 0.0, 1.0)
v_Color in a
fragment shader

Interpolation of color values

CHAPTER 5

Using Colors and Texture Images

The colors generated
by interpolation

In this case, R decreases from 1.0 to 0.0, B increases from 0.0 to 1.0, and all the RGB
values between the two vertices are calculated appropriately—this is called the interpolation process. Once the new color for each of the fragments located between the two vertices is calculated in this way, it is passed to the fragment shader’s v_Color.
We follow an identical procedure in the case of the colored triangle, which is reproduced
in Listing 5.6. After the three vertices’ colors are assigned to the varying variable v_Color
(line 9), the interpolated color for each fragment is passed to the fragment shader’s
v_Color. Once this is assigned to the gl_FragColor at line 19, a colored triangle is drawn,
as shown in Figure 5.8. This interpolation process is carried out for each of the varying
variables. If you want to understand more about this process, a good source of information is the book Computer Graphics.
Listing 5.6 ColoredTriangle.js
1 // ColoredTriangle.js
2 // Vertex shader program
3 var VSHADER_SOURCE = '\
...
6 varying vec4 v_Color;\
7 void main() {\
8
gl_Position = a_Position;\
9
v_Color = a_Color;\ <- The color at line 59 is assigned to v_Color
10 }';
11
12 // Fragment shader program
13 var FSHADER_SOURCE =
...
17 varying vec4 v_Color;\ <- The interpolated color is passed to v_Color
18 void main() {\
19
gl_FragColor = v_Color;\ <- The color is assigned to gl_FragColor
20 }';
21
22 function main() {
...
53
gl.drawArrays(gl.TRIANGLES, 0, n);
54 }
55
56 function initVertexBuffers(gl) {
57
var verticesColors = new Float32Array([
58
// Vertex coordinates and color
59
0.0, 0.5, 1.0, 0.0, 0.0,
60
-0.5, -0.5, 0.0, 1.0, 0.0,
61
0.5, -0.5, 0.0, 0.0, 1.0,
62
]);
...
99 }

Color Triangle (ColoredTriangle.js)

159

In summary, this section has highlighted the critical rasterization process that takes place
between the vertex and fragment shaders. Rasterization is a key component of 3D graphics and is responsible for taking geometric shapes and building up the fragments that will
draw those shapes. After converting the specified geometric shape into fragments (rasterization), it’s possible to set a different color for each of the fragments inside the fragment
shader. This color can be interpolated or set directly by the programmer.

Pasting an Image onto a Rectangle
In the previous section, you explored how to use color when drawing shapes and how
interpolation creates smooth color transitions. Although powerful, this approach is limited
when it comes to reproducing complex visual representations. For example, a problem
arises if you want to create a wall that has the look and feel of the one shown in Figure
5.17, you would need many triangles, and determining the color and coordinates for each
triangle would prove to be daunting.

Figure 5.17

An example of a complex wall surface

As you’d imagine, in 3D graphics, one of the most important processes is actually solving
this problem. The problem is resolved using a technique called texture mapping, which
can re-create the look of real-world materials. The process is actually straightforward and
consists of pasting an image (like a decal) on the surface of a geometrical shape. By pasting
an image from a real-world photograph on a rectangle made up of two triangles, you can
give the rectangle surface an appearance similar to that of a picture. The image is called a
texture image or a texture.
The role of the texture mapping process is to assign the texture image’s pixel colors to the
fragments generated by the rasterization process introduced in the previous section. The
pixels that make up the texture image are called texels (texture elements), and each texel
codes its color information in the RGB or RGBA format (see Figure 5.18).

160

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Each square is a texel

Enlargement

Figure 5.18

Texels

Texture mapping involves the following four steps in WebGL:
1. Prepare the image to be mapped on the geometric shape.
2. Specify the image mapping method for the geometric shape.
3. Load the texture image and configure it for use in WebGL.
4. Extract the texels from the image in the fragment shader, and accordingly set the
corresponding fragment.
To understand the mechanisms involved in texture mapping, let’s examine the sample
program TextureQuad, which “pastes” an image onto a rectangle. If you run it from the
book’s website, you’ll see the result as shown in Figure 5.19 (left).
Note When you want to run the sample programs that use texture images in Chrome
from your local disk, you should add the option --allow-file-access-from-files to
Chrome. This is for security reasons. Chrome, by default, does not allow access to local
files such as ../resources/sky.jpg. For Firefox, the equivalent parameter, set via
about:config, is security.fileuri.strict_origin_policy, which should be set to
false. Remember to set it back when you’re finished because you open a security loophole if local file access is enabled.

Texture image

sky.jpg

Figure 5.19

TextureQuad (left) and the texture image used (right)

Looking in a little more detail at steps (1) to (4) in the following sections, the image
prepared in (1) can be any format that can be displayed in a browser. For now, you can

Pasting an Image onto a Rectangle

161

use any pictures you might have taken yourself or alternatively you can use the images
located in the resource folder of the companion website provided with this book.
The mapping method specified in (2) consists of designating “which part of the texture
image” should be pasted to “which part of the geometric shape”. The part of the geometric shape meant to be covered with the texture is specified using the coordinates of the
vertices that compose a surface. The part of the texture image to be used is specified using
texture coordinates. These are a new form of coordinates so let’s look at how they work.

Texture Coordinates
The texture coordinate system used in WebGL is two-dimensional, as shown in Figure
5.20. To differentiate the texture coordinates from the widely used x and y axis, WebGL
changes the denomination to the s and t coordinates (st coordinates system).2

t
(0.0, 1.0)

(1.0, 1.0)

Texture Image

Text at the texture coordinate
(0.7, 0.4)

s
(0.0, 0.0)

Figure 5.20

(1.0, 0.0)

WebGL’s Texture coordinate system

As you can see from Figure 5.20, the coordinates of the four corners are defined as left
bottom corner (0.0, 0.0), right bottom corner (1.0, 0.0), right top corner (1.0, 1.0), and
left top corner (0.0, 1.0). Because these values are not related to the image size, this allows
a common approach to image handling; for example, whether the texture image’s size is
128×128 or 128×256, the right top corner coordinates will always be
(1.0, 1.0).

Pasting Texture Images onto the Geometric Shape
As previously mentioned, in WebGL, by defining the correspondence between the texture
coordinates and the vertex coordinates of the geometric shape, you can specify how the
texture image will be pasted (see Figure 5.21).

2

The uv coordinates are often used. However, we are using st coordinates because GLSL ES uses the
component names to access the texture image.

162

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Using Colors and Texture Images

t
(0.0, 1.0)

(1.0, 1.0)

y

s
(0.0, 0.0)

(-05, 0.5, 0.0)

(05, 0.5, 0.0)

(1.0, 0.0)

x

Texture coordinate system
(-05, -0.5, 0.0)

(05, -0.5, 0.0)

WebGL coordinate system

Figure 5.21

The resulting image produced by WebGL

Texture coordinates and mapping them to vertices

Here, the texture coordinates (0.0, 1.0) are mapped onto the vertex coordinates (–0.5, 0.5,
0.0), and the texture coordinates (1.0, 1.0) are mapped onto the vertex coordinates (0.5,
0.5, 0.0). By establishing the correspondence for each of the four corners of the texture
image, you obtain the result shown in the right part of Figure 5.21.
Now, given your understanding of how images can be mapped to shapes, let’s look at the
sample program.

Sample Program (TexturedQuad.js)
In TexturedQuad.js (see Listing 5.7), the texture mapping affects both the vertex and the
fragment shaders. This is because it sets the texture coordinates for each vertex and then
applies the corresponding pixel color extracted from the texture image to each fragment.
There are five main parts to the example, each identified by the numbers to the right of
the code.
Listing 5.7
1
2
3
4
5
6
7
8
9
10
11
12
13

TexturedQuad.js

// TexturedQuad.js
// Vertex shader program
var VSHADER_SOURCE =
'attribute vec4 a_Position;\n' +
'attribute vec2 a_TexCoord;\n' +
'varying vec2 v_TexCoord;\n' +
'void main() {\n' +
' gl_Position = a_Position;\n' +
' v_TexCoord = a_TexCoord;\n' +
'}\n';

// Fragment shader program
var FSHADER_SOURCE =
...
17
'uniform sampler2D u_Sampler;\n' +
18
'varying vec2 v_TexCoord;\n' +
19
'void main() {\n' +

<- (Part1)

<- (Part2)

Pasting an Image onto a Rectangle

163

20
' gl_FragColor = texture2D(u_Sampler, v_TexCoord);\n' +
21
'}\n';
22
23 function main() {
...
40
// Set the vertices information
<- (Part3)
41
var n = initVertexBuffers(gl);
...
50
// Setting the textures
51
if (!initTextures(gl, n)) {
...
54
}
55 }
56
57 function initVertexBuffers(gl) {
58
var verticesTexCoords = new Float32Array([
59
// Vertices coordinates, textures coordinates
60
-0.5, 0.5,
0.0, 1.0,
61
-0.5, -0.5,
0.0, 0.0,
62
0.5, 0.5,
1.0, 1.0,
63
0.5, -0.5,
1.0, 0.0,
64
]);
65
var n = 4; // The number of vertices
66
67
// Create the buffer object
68
var vertexTexCoordBuffer = gl.createBuffer();
...
74
// Write the vertex coords and textures coords to the object buffer
75
gl.bindBuffer(gl.ARRAY_BUFFER, vertexTexCoordBuffer);
76
gl.bufferData(gl.ARRAY_BUFFER, verticesTexCoords, gl.STATIC_DRAW);
77
78
var FSIZE = verticesTexCoords.BYTES_PER_ELEMENT;
...
85 gl.vertexAttribPointer(a_Position, 2, gl.FLOAT, false, FSIZE * 4, 0);
86 gl.enableVertexAttribArray(a_Position);
// Enable buffer allocation
87
88
// Allocate the texture coordinates to a_TexCoord, and enable it.
89
var a_TexCoord = gl.getAttribLocation(gl.program, 'a_TexCoord');
...
94
95

gl.vertexAttribPointer(a_TexCoord, 2, gl.FLOAT, false, FSIZE * 4, FSIZE * 2);
gl.enableVertexAttribArray(a_TexCoord); // Enable buffer allocation
...
97
return n;
98 }
99

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100 function initTextures(gl, n)
<- (Part4)
101
var texture = gl.createTexture();
// Create a texture object
...
107
// Get the storage location of the u_Sampler
108
var u_Sampler = gl.getUniformLocation(gl.program, 'u_Sampler');
...
114
var image = new Image(); // Create an image object
...
119
// Register the event handler to be called on loading an image
120
image.onload = function(){ loadTexture(gl, n, texture, u_Sampler, image); };
121
// Tell the browser to load an image
122
image.src = '../resources/sky.jpg';
123
124
return true;
125 }
126
127 function loadTexture(gl, n, texture, u_Sampler, image){
<- (Part5)
128
gl.pixelStorei(gl.UNPACK_FLIP_Y_WEBGL, 1); // Flip the image's y axis
129
// Enable the texture unit 0
130
gl.activeTexture(gl.TEXTURE0);
131
// Bind the texture object to the target
132
gl.bindTexture(gl.TEXTURE_2D, texture);
133
134
// Set the texture parameters
135
gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_MIN_FILTER, gl.LINEAR);
136
// Set the texture image
137
gl.texImage2D(gl.TEXTURE_2D, 0, gl.RGB, gl.RGB, gl.UNSIGNED_BYTE, image);
138
139
// Set the texture unit 0 to the sampler
140
gl.uniform1i(u_Sampler, 0);
...
144
gl.drawArrays(gl.TRIANGLE_STRIP, 0, n); // Draw a rectangle
145 }

This program is structured into five main parts:
Part 1: Receive the texture coordinates in the vertex shader and then pass them to the
fragment shader.
Part 2: Paste the texture image onto the geometric shape inside the fragment shader.
Part 3: Set the texture coordinates (initVertexBuffers()).
Part 4: Prepare the texture image for loading, and request the browser to read it.
(initTextures()).
Part 5: Configure the loaded texture so that it can be used in WebGL (loadTexture()).

Pasting an Image onto a Rectangle

165

Let’s look at the sequence starting from Part 3: the process to set the texture coordinates
using initVertexBuffers(). The shaders are executed after loading the image, so we will
explain them at the end.

Using Texture Coordinates (initVertexBuffers())
You pass texture coordinates to the vertex shader using the same approach you’ve been
using to pass other vertex data to the vertex shader, by combining vertex coordinates and
vertex data into a single buffer. Line 58 defines an array verticesTexCoords containing
pairs of vertex coordinates and their associated texture coordinates:
58
59
60
61
62
63
64

var verticesTexCoords = new Float32Array([
// Vertex coordinates and texture coordinates
-0.5,
0.5, 0.0, 1.0,
-0.5, -0.5, 0.0, 0.0,
0.5,
0.5, 1.0, 1.0,
0.5, -0.5, 1.0, 0.0,
]);

As you can see, the first vertex (–0.5, 0.5) is mapped to the texture coordinate (0.0, 1.0),
the second vertex (–0.5, –0.5) is mapped to the texture coordinate (0.0, 0.0), the third
vertex (0.5, 0.5) is mapped to the texture coordinate (1.0, 10), and the fourth vertex (0.5,
–0.5) is mapped to the texture coordinate (1.0, 0.0). Figure 5.21 illustrates these mappings.
Lines 75 to 86 then write vertex coordinates and texture coordinates to the buffer object,
assign it to a_Position, and enable the assignment. After that, lines 89 to 94 retrieve the
storage location of the attribute variable a_TexCoord and then assign the buffer object
containing the texture coordinates to the variable. Finally, line 95 enables the assignment
of the buffer object to a_TexCoord:
88
89
94

// Assign the texture coordinates to a_TexCoord, and enable it.
var a_TexCoord = gl.getAttribLocation(gl.program, 'a_TexCoord');
...
gl.vertexAttribPointer(a_TexCoord, 2, gl.FLOAT, false, FSIZE * 4,

95

gl.enableVertexAttribArray(a_TexCoord);

➥FSIZE * 2);

Setting Up and Loading Images (initTextures())
This process is performed from lines 101 to 122 in initTextures(). Line 101 creates
a texture object (gl.createTexture()) for managing the texture image in the WebGL
system, and line 108 gets the storage location of a uniform variable (gl.getUniformLocation()) to pass the texture image to the fragment shader():
101

var texture = gl.createTexture(); // Create a texture object
...
108
var u_Sampler = gl.getUniformLocation(gl.program, 'u_Sampler');

166

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A texture object is created using gl.createTexture().

gl.createTexture()
Create a texture object to hold a texture image.
Parameters

None

Return value

non-null

The newly created texture object.

null

Failed to create a texture object.

Errors

None

This call creates the texture object in the WebGL system, as shown in Figure 5.22.
gl.TEXTURE0 to gl.TEXTURE7 are texture units for managing a texture image, and each has
an associated gl.TEXTURE_2D, which is the texture target for specifying the type of texture.

This will be explained in detail later.
uniform variable

JavaScript

g l.TEXTURE0

g l.TEXTURE1 …

g l.TEXTURE7

…

fu n ctio n m a in () {
v a r g l= g e tW e b G L …
…

g l.TEXTURE_2D

g l.TEXTURE_2D

g l.TEXTURE_2D

g l.TEXTURE0

g l.TEXTURE1 …

g l.TEXTURE7

F ra g m e n t
S hader

uniform variable

JavaScript

…

fu n ctio n m a in () {
v a r g l= g e tW e b G L …
…
gl.createTexture( … );

Figure 5.22

g l.TEXTURE_2D

g l.TEXTURE_2D

g l.TEXTURE_2D

F ra g m e n t
S hader

Texture
Object

Create a texture object

The texture object can be deleted using gl.deleteTexture(). Note, if this method is called
with a texture object that has already been deleted, the call has no effect.
gl.deleteTexture(texture)
Delete the texture object specified by texture.
Parameter

texture

Return value

None

Errors

None

Specifies the texture object to be deleted.

Pasting an Image onto a Rectangle

167

In the next step, it’s necessary to request that the browser load the image that will be
mapped to the rectangle. You need to use an Image object for this purpose:
114
119
120
121
122

var image = new Image(); // Create an Image object
...
// Register an event handler to be called when image loading completes
image.onload = function(){ loadTexture(gl, n, texture, u_Sampler, image); };
// Tell the browser to load an image
image.src = '../resources/sky.jpg';

This code snippet creates an Image object, registers the event handler (loadTexture()) to
be called on loading the image, and tells the browser to load the image.
You need to create an Image object (a special JavaScript object that handles images) using
the new operator, just as you would do for an Array object or Date object. This is done at
line 114.
114

var image = new Image();

// Create an Image object

Because loading of images is performed asynchronously (see the boxed article that
follows), when the browser signals completion of loading, it needs to pass the image
to the WebGL system. Line 120 handles this, telling the browser that, after loading the
image, the anonymous function loadTexture() should be called.
120

image.onload = function(){ loadTexture(gl, n, texture, u_Sampler, image); };

loadTexture() takes five parameters, with the newly loaded image being passed via
the variable (Image object) as the last argument image. gl is the rendering context for
WebGL, n is the number of vertices, texture is the texture object created at line 101, and
u_Sampler is the storage location of a uniform variable.

Just like the  tag in HTML, we can tell the browser to load the texture image by
setting the image filename to the property src of the Image object (line 122). Note that
WebGL, because of the usual browser security restrictions, is not allowed to use images
located in other domains for texture images:
122

image.src = '../resources/sky.jpg';

After executing line 122, the browser starts to load the image asynchronously, so the
program continues on to the return statement at line 124 and then exits. When the
browser finishes loading the image and wants to pass the image to the WebGL system, the
event handler loadTexture() is called.

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Asynchronous Loading Texture Images
Usually, OpenGL applications written in C or C++ load the texture image files straight
from the hard disk where they are stored. However, because WebGL programs are
running inside the browser, it is impossible to load images directly. Instead, it is
necessary to read images indirectly by requesting the browser to do it. (Typically, the
browser sends a request to the web server to obtain the image.) The advantage is that
you can use any kind of image a browser can display, but it also makes the process
more complex because you now have to handle two processes (the browser loading
request, and the actual WebGL loading) that behave “asynchronously” (they run in the
background) and thus do not block execution of the program.
Figure 5.23 shows the substeps between [1] tell the browser to load an image and [7] call
the function loadTexture() after completing loading the texture image.
[1] Tell the browser to call
loadTexture() after loading
an image.

[3] Request the
Web server to
retrieve an
image.

JavaScript
initTextures() {
…
// Register the event on loading an image.
image.onload = function(){ loadTexture(…); };
// Tell the browser to load an image.
image.src = '../resources/sky.jpg';
…
}
[2] Tell the browser to
function loadTexture(…) {
…
}

[4] Load image.
[5]
Complete
retrieving an
image.

load an image.

Browser
[7] Call loadTexture().

Figure 5.23

Web
server

[6] Complete
loading an image.

Image file
(sky.jpg)

Asynchronous loading texture images

In Figure 5.23, [1] and [2] are executed sequentially, but [2] and [7] are not. After
requesting the browser to load an image in [2], the JavaScript program doesn’t wait for
the image to be loaded, but proceeds to the next stage. (This behavior will be explained
in detail in a moment.) While the JavaScript program is continuing, the browser
sends a request to the web server for the image [3]. When the image loading process is
completed [4] and [5], the browser tells the JavaScript program that the image loading
has completed [6]. This kind of behavior is referred to as asynchronous.
The image loading process is analogous to the way a web page written in HTML displays
images. In HTML, an image is displayed by specifying the file URL to the src attribute of
the  tag (below) causing the browser to load the image from the specified URL. This
part corresponds to [2] shown in Figure 5.23.


Pasting an Image onto a Rectangle

169

The asynchronous nature of the image loading process can be easily understood by
considering how a web page that includes numerous images is displayed. Typically,
the page text and layout are displayed rapidly, and then images appear slowly as
they are loaded. This is because the image loading and display processes are executed
asynchronously, allowing you to view and interact with the web page without having to
wait for all images to load.

Make the Texture Ready to Use in the WebGL System (loadTexture())
The function loadTexture() is defined as follows:
127 function loadTexture(gl, n, texture, u_Sampler, image) { <- (Part5)
128
gl.pixelStorei(gl.UNPACK_FLIP_Y_WEBGL, 1); // Flip the image's y axis
129
// Enable the texture unit 0
130
gl.activeTexture(gl.TEXTURE0);
131
// Bind the texture object to the target
132
gl.bindTexture(gl.TEXTURE_2D, texture);
133
134
// Set the texture parameters
135
gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_MIN_FILTER, gl.LINEAR);
136
// Set the texture image
137
gl.texImage2D(gl.TEXTURE_2D, 0, gl.RGB, gl.RGB, gl.UNSIGNED_BYTE, image);
138
139
// Set the texture unit 0 to the sampler
140
gl.uniform1i(u_Sampler, 0);
...
144
gl.drawArrays(gl.TRIANGLE_STRIP, 0, n); // Draw a rectangle
145 }

Its main purpose is to prepare the image for use by the WebGL system, which it does
using a texture object that is prepared and used in a similar manner to a buffer object.
The following sections explain the code in more detail.

Flip an Image’s Y-Axis
Before using the loaded images as a texture, you need to flip the y-axis:
128

gl.pixelStorei(gl.UNPACK_FLIP_Y_WEBGL, 1);// Flip the image's y-axis

This method flips an image’s Y-axis when it’s loaded. As shown in Figure 5.24, the t-axis
direction of the WebGL texture coordinate system is the inverse of the y-axis direction of
the coordinate system used by PNG, BMP, JPG, and so on. For this reason, if you flip the
image’s Y-axis, you can map the image to the shape correctly. (You could also flip the t
coordinates by hand instead of flipping the image.)

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t
x

s

y
Image coordinate system
Figure 5.24

WebGL coordinate system

The image coordinate system and WebGL texture coordinate system

The following explains gl.pixelStorei().

gl.pixelStorei(pname, param)
Perform the process defined by pname and param after loading an image.
Parameters

pname

Specifies any of the following:

gl.UNPACK_FLIP_Y_WEBGL

Flips an image’s Y-axis after loading the
image. The default value is false.

gl.UNPACK_PREMULTIPLY_
ALPHA_WEBGL

Multiplies each component of RGB in an
image by A in the image. The default value is
false.

param

Return value

None

Errors

INVALID_ENUM

Specifies none-zero (means true) or zero
(means false). It must be specified in the
integer.

pname is none of these values.

Making a Texture Unit Active (gl.activeTexture())
WebGL supports multiple texture images (multitexturing) using a mechanism called
a texture unit. A texture unit manages texture images by using a unit number for each
texture. Because of this, even if you only want to use a single texture image, you must
specify and use a texture unit.

Pasting an Image onto a Rectangle

171

The number of texture units supported varies according to your hardware and WebGL
implementation, but by default at least eight texture units are supported, and some
systems will support more. The built-in constants, gl.TEXTURE0, gl.TEXTURE1, ..., and
gl.TEXTURE7, represent each texture unit (see Figure 5.25).
Texture unit

uniform variable

JavaS cript
fu n c tio n m a in ( ) {
v a r g l= g e tW e b G L …
…

Figure 5.25

gl.TEXTURE0

gl.TEXTURE1 …

gl.TEXTURE7

gl.TEXTURE_2D

gl.TEXTURE_2D

gl.TEXTURE_2D

F ra g m e n t
Shader

Multiple texture units managed by WebGL

Before using a texture unit, it must be made active using a call to gl.activeTexture() (see
Figure 5.26):
132
133

// Make the texture unit 0 active
gl.activeTexture(gl.TEXTURE0);

gl.activeTexture(texUnit)
Make the texture unit specified by texUnit active.
Parameters

texUnit

Specifies the texture unit to be made active: gl.TEXTURE0,
gl.TEXTURE1, ..., or gl.TEXTURE7. The tailing number indicates the texture unit number.

Return value

None

Errors

INVALID_ENUM:

uniform variable

JavaS cript
fu n c tio n m a in ( ) {
v a r g l= g e tW e b G L …
…
gl.activeTexture(…);

Figure 5.26

172

texUnit is none of these values

gl.TEXTURE0

gl.TEXTURE1 …

gl.TEXTURE7

gl.TEXTURE_2D

gl.TEXTURE_2D

gl.TEXTURE_2D

T e x tu re
O b je c t

Activate texture unit (gl.TEXTURE0)

CHAPTER 5

Using Colors and Texture Images

F ra g m e n t
Shader

Binding a Texture Object to a Target (gl.bindTexture())
Next, you need to tell the WebGL system what types of texture image is used in the
texture object. You do this by binding the texture object to the target in a similar way to
that of the buffer objects explained in the previous chapter. WebGL supports two types of
textures, as shown in Table 5.2.
Table 5.2

Types of Textures

Type of Texture

Description

gl.TEXTURE_2D

Two-dimensional texture

gl.TEXTURE_CUBE_MAP Cube map texture

The sample program uses a two-dimensional image as a texture and specifies gl.
TEXTURE_2D at line 132. The cube map texture is beyond the scope of this book. If you are
interested in more information, please refer to the book OpenGL ES 2.0 Programming Guide:
131
132

// Bind the texture object to the target
gl.bindTexture(gl.TEXTURE_2D, texture);

gl.bindTexture(target, texture)
Enable the texture object specified by texture and bind it to the target. In addition, if a
texture unit was made active by gl.activeTexture(), the texture object is also bound to
the texture unit.
Parameters

target

Specifies gl.TEXTURE_2D or gl.TEXTURE_CUBE_MAP.

texture

Specifies the texture object to be bound.

Return value

None

Errors

INVALID_ENUM target is none of these values.

Note that this method performs two tasks: enabling the texture object and binding it
to target, and binding it to the texture unit. In this case, because the texture unit 0
(gl.TEXTURE0) is active, after executing line 136, the internal state of the WebGL
system is changed, as shown in Figure 5.27.

Pasting an Image onto a Rectangle

173

uniform variable

JavaS cript
fu n c tio n m a in ( ) {
v a r g l= g e tW e b G L …
…
g l.b in d T e x tu r e ( … ) ;

Figure 5.27

g l.TEXTURE0

g l.TEXTURE1 …

g l.TEXTURE7

g l.TEXTURE_2D

g l.TEXTURE_2D

g l.TEXTURE_2D

F ra g m e n t
Shader

Texture
Object

Bind a texture object to the target

At this stage, the program has specified the type of texture that is used in the texture
object (gl.TEXTURE_2D) and that will be used to deal with the texture object in the future.
This is important, because in WebGL, you cannot manipulate the texture object directly.
You need to do that through the binding.

Set the Texture Parameters of a Texture Object (gl.texParameteri())
In the next step, you need to set the parameters (texture parameter) that specify how the
texture image will be processed when the texture image is mapped to shapes. The generic
function gl.texParameteri() can be used to set texture parameters.

gl.texParameteri(target, pname, param)
Set the value specified by param to the texture parameter specified by pname in the
texture object bound to target.
Parameters

target

Specifies gl.TEXTURE_2D or gl.TEXTURE_CUBE_MAP.

pname

Specifies the name of the texture parameter (Table 5.3).

param

Specifies the value set to the texture parameter pname
(Table 5.4, Table 5.5).

Return value None
Errors

INVALID_ENUM

target is none of the preceding values

INVALID_OPERATION

no texture object is bound to target

There are four texture parameters available, illustrated in Figure 5.28, which you can
specify to pname:

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• Magnification method (gl.TEXTURE_MAG_FILTER): The method to magnify a texture
image when you map the texture to a shape whose drawing area is larger than the
size of the texture. For example, when you map a 16×16 pixel image to a 32×32 pixel
shape, the texture should be doubled in size. WebGL needs to fill the gap between
texels due to the magnification, and this parameter specifies the method used to fill
the gap.

• Minification method (gl.TEXTURE_MIN_FILTER): The method of minifying a texture
image when you map the texture to a shape whose drawing area is smaller than
the size of the texture. For example, when you map a 32×32 pixel image to a 16×16
pixel shape, the texture should be reduced in size. To do that, the system needs to
cull texels to fit the target size. This parameter specifies the method used to cull
texels.

• Wrapping method on the left and right side (gl.TEXTURE_WRAP_S): How to fill the
remaining regions on the left side and the right side of a subregion when you map a
texture image to the subregion of a shape.

• Wrapping method on top and bottom (gl.TEXTURE_WRAP_T): Similar to (3), the
method used to fill the remaining regions in the top and bottom of a subregion.

Minification

Magnification

gl.TEXTURE_MAG_FILTER

gl.TEXTURE_MIN_FILTER

The method to fill this region.

gl.TEXTURE_WRAP_S

The method to fill this region.

gl.TEXTURE_WRAP_T

Figure 5.28

Four texture parameters and their effects

Table 5.3 shows each texture parameter and its default value.
Pasting an Image onto a Rectangle

175

Table 5.3

Texture Parameters and Their Default Values

Texture Parameter

Description

Default Value

gl.TEXTURE_MAG_FILTER

Texture magnification

gl.LINEAR

gl.TEXTURE_MIN_FILTER

Texture minification

gl.NEAREST_MIPMAP_LINEAR

gl.TEXTURE_WRAP_S

Texture wrapping in s-axis

gl.REPEAT

gl.TEXTURE_WRAP_T

Texture wrapping in t-axis

gl.REPEAT

We also show the constant values that can be specified to gl.TEXTURE_MAG_FILTER and
gl.TEXTURE_MIN_FILTER in Table 5.4 and gl.TEXTURE_WRAP_S and gl.TEXTURE_WRAP_T in
Table 5.5.
Table 5.4 Non-Mipmapped Values, Which Can be Specified to gl.TEXTURE_MAG_FILTER and
gl.TEXTURE_MIN_FILTER3
Value

Description

gl.NEAREST

Uses the value of the texel that is nearest (in Manhattan distance) the center
of the pixel being textured.

gl.LINEAR

Uses the weighted average of the four texels that are nearest the center
of the pixel being textured. (The quality of the result is clearer than that of
gl.NEAREST, but it takes more time.)

Table 5.5

Values that Can be Specified to gl.TEXTURE_WRAP_S and gl.TEXTURE_WRAP_T

Value

Description

gl.REPEAT

Use a texture image repeatedly

gl.MIRRORED_REPEAT

Use a texture image mirrored-repeatedly

gl.CLAMP_TO_EDGE

Use the edge color of a texture image

As shown in Table 5.3, each parameter has a default value, and you can generally use the
default value as is. However, the default value of gl.TEXTURE_MIN_FILTER is for a special
texture format called MIPMAP. A MIPMAP is a sequence of textures, each of which is
a progressively lower resolution representation of the same image. Because a MIPMAP
texture is not often used, we don’t cover it in this book. For this reason, you set the value
gl.LINEAR to the texture parameter gl.TEXTURE_MIN_FILTER at line 135:
3

Although omitted in this table, other values can be specified for a MIPMAP texture: gl.NEAREST_
MIPMAP_NEAREST, gl.LINEAR_MIPMAP_NEAREST, gl.NEAREST_MIPMAP_LINEAR, and gl.LINEAR_
MIPMAP_LINEAR. See the book OpenGL Programming Guide for these values.

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134
135

// Set the texture parameters
gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_MIN_FILTER, gl.LINEAR);

After executing line 139, the value is set to the texture object, and then the internal state
of the WebGL system is modified as shown in Figure 5.29.
uniform variable

JavaS cript
fu n c tio n m a in ( ) {
v a r g l= g e tW e b G L …
…
g l.te x P a r a m e te r i( … ) ;

g l.TEXTURE0

g l.TEXTURE1 …

g l.TEXTURE7

g l .TEXTURE_2D

g l.TEXTURE_2D

g l.TEXTURE_2D

F ra g m e n t
Shader

Texture Object
g l .T E X T U R E _ M I N _ F I L T E R
g l.L IN E A R

Figure 5.29

Set texture parameter

The next step is to assign a texture image to the texture object.

Assigning a Texture Image to a Texture Object (gl.texImage2D())
To assign an image to a texture object, you use the method gl.texImage2D(). In addition
to assigning a texture, this method allows you to tell the WebGL system about the image
characteristics.

gl.texImage2D(target, level, internalformat, format, type, image)
Set the image specified by image to the texture object bound to target.
Parameters

target

Specifies gl.TEXTURE_2D or gl.TEXTURE_CUBE_MAP.

level

Specified as 0. (Actually, this parameter is used for a
MIPMAP texture, which is not covered in this book.)

internalformat

Specifies the internal format of the image (Table 5.6).

format

Specifies the format of the texel data. This must be
specified using the same value as internalformat.

type

Specifies the data type of the texel data (Table 5.7).

image

Specifies an Image object containing an image to be
used as a texture.

Return value

None

Errors

INVALID_ENUM

target is none of the above values.

INVALID_OPERATION

No texture object is bound to target

Pasting an Image onto a Rectangle

177

This method is used at line 136 in the sample program:
137 gl.texImage2D(gl.TEXTURE_2D, 0, gl.RGBA, gl.RGBA, gl.UNSIGNED_BYTE, image);

After executing line 144, the texture image loaded into the Image object image in
JavaScript is passed to the WebGL system (see Figure 5.30).
uniform variable

JavaS cript
fu n c tio n m a in ( ) {
v a r g l= g e tW e b G L …
…
g l.te x Im a g e 2 D ( … ) ;

g l.TEXTURE0

g l.TEXTURE1 …

g l.TEXTURE7

g l.TEXTURE_2D

g l.TEXTURE_2D

g l.TEXTURE_2D

image

F ra g m e n t
Shader

Texture Object
gl.TEXTURE_MIN_FILTER
gl.LINEAR

Figure 5.30

Assign an image to the texture object

Let’s take a quick look at each parameter of this method. You must specify 0 to level
because you aren’t using a MIPMAP texture. The format specifies the format of the texel
data, with available formats shown in Table 5.6. You need to select an appropriate format
for the image used as a texture. The sample program uses gl.RGB format because it uses a
JPG image in which each pixel is composed of RGB components. For other formats, such
as PNG, images are usually specified as gl.RGBA, and BMP images are usually specified as
gl.RGB. gl.LUMINANCE and gl.LUMINANCE_ALPHA are used for a grayscale image and so on.
Table 5.6

The Format of the Texel Data

Format

Components in a Texel

gl.RGB

Red, green, blue

gl.RGBA

Red, green, blue, alpha

gl.ALPHA

(0.0, 0.0, 0.0, alpha)

gl.LUMINANCE

L, L, L, 1 L: Luminance

gl.LUMINANCE_ALPHA

L, L. L. alpha

Here, luminance is the perceived brightness of a surface. It is often calculated as a
weighted average of red, green, and blue color values that gives the perceived brightness
of the surface.
As shown in Figure 5.30, this method stores the image in the texture object in the WebGL
system. Once stored, you must tell the system about the type of format the image uses

178

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Using Colors and Texture Images

using the internalformat parameter. As mentioned, in WebGL, internalformat must specify
the same value as format.
The type specifies the data type of the texel data (see Table 5.7). Usually, we specify gl.
UNSIGNED_BYTE as the data type. Other data types are also available, such as gl.UNSIGNED_
SHORT_5_6_5 (which packs RGB components into 16 bits). These types are used for passing
compressed images to the WebGL system to reduce loading time.
Table 5.7

The Data Type of Texel Data

Type

Description

gl.UNSIGNED_BYTE

Unsigned byte format. Each color component has 1 byte.

gl.UNSIGNED_SHORT_5_6_5

RGB: Each component has 5, 6, and 5 bits, respectively.

gl.UNSIGNED_SHORT_4_4_4_4

RGBA: Each component has 4, 4, 4, and 4 bits, respectively.

gl.UNSIGNED_SHORT_5_5_5_1

RGBA: Each RGB component has 5 bits, and A has 1 bit.

Pass the Texture Unit to the Fragment Shader (gl.uniform1i())
Once the texture image has been passed to the WebGL system, it must be passed to the
fragment shader to map it to the surface of the shape. As explained before, a uniform variable is used for this purpose because the texture image does not change for each fragment:
13 var FSHADER_SOURCE =
...
17
'uniform sampler2D u_Sampler;\n' +
18
'varying vec2 v_TexCoord;\n' +
19
'void main() {\n' +
20
' gl_FragColor = texture2D(u_Sampler, v_TexCoord);\n' +
21
'}\n';

This uniform variable must be declared using the special data type for textures shown in
Table 5.8. The sample program uses a two-dimensional texture (gl.TEXTURE_2D), so the
data type is set to sampler2D.
Table 5.8

Special Data Types for Accessing a Texture

Type

Description

sampler2D

Data type for accessing the texture bound to gl.TEXTURE_2D

samplerCube

Data type for accessing the texture bound to gl.TEXTURE_CUBE_MAP

The call to initTextures() (line 100) gets the storage location of this uniform variable
u_Sampler at line 108 and then passes it to loadTexture() as an argument. The uniform

Pasting an Image onto a Rectangle

179

variable u_Sampler is set at line 139 by specifying the texture unit number (“n” in
gl.TEXTUREn) of the texture unit that manages this texture object. In this sample program,
you specify 0 because you are using the texture object bound to gl.TEXTURE0 in the call to
gl.uniformi():
138
139

// Set the texture unit 0 to the sampler
gl.uniform1i(u_Sampler, 0);

After executing line 139, the WebGL system is modified as shown in Figure 5.31, thereby
allowing access to the image in the texture object from the fragment shader.
u _ S a m p le r

JavaS cript
fu n c tio n m a in ( ) {
v a r g l= g e tW e b G L …
…
g l.te x Im a g e 2 D ( … ) ;

g l.TEXTURE0

g l.TEXTURE1 …

g l.TEXTURE7

g l .TEXTURE_2D

g l.TEXTURE_2D

g l.TEXTURE_2D

image

F ra g m e n t
S h ad er

Texture Object
gl.TEXTURE_MIN_FILTER
gl.LINEAR

Figure 5.31

Set texture unit to uniform variable

Passing Texture Coordinates from the Vertex Shader to the Fragment
Shader
Because the texture coordinates for each vertex are passed to the attribute variable
a_TexCoord, it’s possible to pass the data to the fragment shader through the varying
variable v_TexCoord. Remember that varying variables of the same name and type are
automatically copied between the vertex shader and the fragment shader. The texture
coordinates are interpolated between vertices, so you can use the interpolated texture
coordinates in a fragment shader to specify each texture coordinate for each fragment:
2 // Vertex shader program
3 var VSHADER_SOURCE =
4
'attribute vec4 a_Position;\n' +
5
'attribute vec2 a_TexCoord;\n' +
6
'varying vec2 v_TexCoord;\n' +
7
'void main() {\n' +
8
' gl_Position = a_Position;\n' +
9
' v_TexCoord = a_TexCoord;\n' +
10
'}\n';

At this stage, you have completed the preparations for using the texture image in the
WebGL system.

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All that’s left is to read the color of the texel located at the corresponding texture coordinates from the texture image and then use it to set the color of the fragment.

Retrieve the Texel Color in a Fragment Shader (texture2D())
Retrieving a color of a texel from the texture image is done at line 20 in the fragment
shader:
20

'

gl_FragColor = texture2D(u_Sampler, v_TexCoord);\n' +

It uses the GLSL ES built-in function texture2D() to read out the texel color from the
shader. texture2D() is an easy-to-use function that can retrieve the texel color at a texture
coordinate by specifying the texture unit number in the first parameter and the texture
coordinates in the second parameter. However, because this function is a built-in function
of GLSL ES, note the data type of the parameters and the return value of the function.

vec4 texture2D(sampler2D sampler, vec2 coord)
Retrieve a texel color at the texture coordinates specified by coord from the texture image
specified by sampler.
Parameters

sampler Specifies the texture unit number.
coord

Return value

Table 5.9

Specifies the texture coordinates.

The texel color (vec4) for the coordinates. The color format changes according to the internalformat specified by gl.texImage2D(). Table 5.9 shows
the differences. If the texture image is not available for some reason, this
function returns (0.0, 0.0, 0.0, 1.0).

Return Value of texture2D()

Internalformat

Return Value

gl.RGB

(R, G, B, 1.0)

gl.RGBA

(R, G, B, A)

gl.ALPHA

(0.0, 0.0, 0.0, A)

gl.LUMINANCE

(L, L, L, 1.0)

gl.LUMINANCE_ALPHA

(L, L, L, A)

L indicates luminance

The texture magnification and minification parameters determine the return value in cases
where WebGL interpolates the texel. Once this function executes, by assigning the return
value to gl_FragColor, the fragment is displayed using the color. As a result of this operation, the texture image is mapped to the shape to be drawn (in this case, a rectangle).
Pasting an Image onto a Rectangle

181

This is the final step in the process needed for texture mapping. At this stage, your texture
image has been loaded, set up in the WebGL system and mapped to the shape you are
drawing.
As you have seen, texture mapping in WebGL seems a complex process partly because it
must deal with an image and request the browser to load it, and partly because you are
required to use the texture unit even if you use only a single texture. However, once you
master the basic steps, they are the same each time you want to map a texture.
The next section explores the use of textures and will familiarize you with the whole
process.

Experimenting with the Sample Program
To familiarize you with texture mapping, let’s modify the sample program by changing
the texture coordinates. For example, modify the texture coordinates in TexturedQuad as
follows:
var verticesTexCoords = new Float32Array([
// Vertex coordinates and texture coordinates
-0.5, 0.5,
-0.3, 1.7,
-0.5, -0.5,
-0.3, -0.2,
0.5, 0.5,
1.7, 1.7,
0.5, -0.5,
1.7, -0.2
]);

If you load the modified program TexturedQuad_Repeat, you’ll see an effect like the
screenshot of Figure 5.32 (left side). To understand what’s happening here, take a look at
the figure on the right side, which shows each texture coordinate in the texture coordinate system.
t
1.7

-0.3

Figure 5.32

(0.0,1.0)

(1.0,1.0)

(0.0,0.0)

(1.0,0.0)

-0.2

1.7

s

Modify the texture coordinate (a screenshot of TexturedQuad_Repeat)

The image isn’t sufficient to cover the larger shape, so as you can see, the texture image is
being repeated. This is driven by the value of gl.TEXTURE_WRAP_S and gl.TEXTURE_WRAP_T,
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Using Colors and Texture Images

which are set to gl.REPEAT in the sample program, telling the WebGL system to repeat the
texture image to fill the area.
Now let’s modify the texture parameters as follows to see what other effects we can
achieve. The modified program is saved as TexturedQuad_Clamp_Mirror, and Figure 5.33
shows the result when run in your browser:
// Set texture parameters
gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_MIN_FILTER, gl.LINEAR);
gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_WRAP_S, gl.CLAMP_TO_EDGE);
gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_WRAP_T, gl.MIRRORED_REPEAT);

You can see that the edge color of the texture is repeated in the s-axis (horizontal axis),
and the texture image itself is mirrored-repeated in the t-axis (vertical axis).

Figure 5.33

TexturedQuad_Clamp_Mirror

That concludes the explanation of the basic texture mapping technique available in
WebGL. The next section builds on this basic technique and explores texture mapping
using multiple texture images.

Pasting Multiple Textures to a Shape
Earlier in the chapter, you learned that WebGL can deal with multiple texture images,
which was the reason for the multiple texture units. The examples so far have used
only one texture, and thus one element of the unit. This section will construct a sample
program, MultiTexture, which pastes two texture images to a rectangle, allowing a
better examination of the texture unit mechanism. Figure 5.34 shows a screenshot of
MultiTexture. As you can see, the two texture images are “blended” to create the composite in the figure.
Pasting Multiple Textures to a Shape

183

Figure 5.34

MultiTexture

Figure 5.35 shows the two separate texture images used in this sample program. To highlight WebGL’s ability to deal with various image formats, the sample program intentionally uses different image formats for each file.

Figure 5.35

Texture images (sky.jpg on left; circle.gif on right) used in MultiTexture

Essentially, you can map multiple texture images to a shape by repeating the process of
mapping a single texture image to a shape described in the previous section. Let’s examine
the sample program to see how that is done.

Sample Program (MultiTexture.js)
Listing 5.8 shows the basic processing flow of MultiTexture.js, which is similar to
TexturedQuad.js with three key differences: (1) the fragment shader accesses two textures,
(2) the final fragment color is calculated from the two texels from both textures, and (3)
initTextures() creates two texture objects.

184

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Listing 5.8

MultiTexture.js

1 // TexturedQuad.js
...
13 var FSHADER_SOURCE =
...
17
'uniform sampler2D u_Sampler0;\n' +
18
'uniform sampler2D u_Sampler1;\n' +
19
'varying vec2 v_TexCoord;\n' +
20
'void main() {\n' +
21
' vec4 color0 = texture2D(u_Sampler0, v_TexCoord);\n' +
22
' vec4 color1 = texture2D(u_Sampler1, v_TexCoord);\n' +
23
' gl_FragColor = color0 * color1;\n' +
24
'}\n';
25
26 function main() {
...
53
// Set textures
54
if (!initTextures(gl, n)) {
...
58 }
59
60 function initVertexBuffers(gl) {
61
var verticesTexCoords = new Float32Array([
62
// Vertex coordinates and texture coordinates
63
-0.5, -0.5,
0.0, 1.0,
64
-0.5, -0.5,
0.0, 0.0,
65
0.5, -0.5,
1.0, 1.0,
66
0.5, -0.5,
1.0, 0.0,
67
]);
68
var n = 4; // The number of vertices
...
100
return n;
101 }
102
103 function initTextures(gl, n) {
104
// Create a texture object
105
var texture0 = gl.createTexture();
106
var texture1 = gl.createTexture();
...
112
// Get the storage locations of u_Sampler1 and u_Sampler2
113
var u_Sampler0 = gl.getUniformLocation(gl.program, 'u_Sampler0');
114
var u_Sampler1 = gl.getUniformLocation(gl.program, 'u_Sampler1');
...
120
// Create Image objects

Pasting Multiple Textures to a Shape

<-(1)
<-(2)

<-(3)

185

121
122
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156

var image0 = new Image();
var image1 = new Image();
...
// Register the event handler to be called when image loading is completed
image0.onload = function(){ loadTexture(gl, n, texture0, u_Sampler0,
➥image0, 0); };
image1.onload = function(){ loadTexture(gl, n, texture1, u_Sampler1,
➥image1, 1); };
// Tell the browser to load an Image
image0.src = '../resources/redflower.jpg';
image1.src = '../resources/circle.gif';
return true;
}
// Specify whether the texture unit is ready to use
var g_texUnit0 = false, g_texUnit1 = false;
function loadTexture(gl, n, texture, u_Sampler, image, texUnit) {
gl.pixelStorei(gl.UNPACK_FLIP_Y_WEBGL, 1);// Flip the image's y-axis
// Make the texture unit active
if (texUnit == 0) {
gl.activeTexture(gl.TEXTURE0);
g_texUnit0 = true;
} else {
gl.activeTexture(gl.TEXTURE1);
g_texUnit1 = true;
}
// Bind the texture object to the target
gl.bindTexture(gl.TEXTURE_2D, texture);

// Set texture parameters
gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_MIN_FILTER, gl.LINEAR);
// Set the texture image
gl.texImage2D(gl.TEXTURE_2D, 0, gl.RGBA, gl.RGBA, gl.UNSIGNED_BYTE, image);
// Set the texture unit number to the sampler
gl.uniform1i(u_Sampler, texUnit);
...
161
if (g_texUnit0 && g_texUnit1) {
162
gl.drawArrays(gl.TRIANGLE_STRIP, 0, n); // Draw a rectangle
163
}
164 }

First, let’s examine the fragment shader. In TexturedQuad.js, because the fragment shader
used only one texture image, it prepared a single uniform variable, u_Sampler. However,
this sample program uses two texture images and needs to define two sampler variables as
follows:
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Using Colors and Texture Images

17
18

'uniform sampler2D u_Sampler0;\n' +
'uniform sampler2D u_Sampler1;\n' +

main() in the fragment shader fetches the texel value from each texture image at lines 21
and 22, storing them to the variables color0 and color1, respectively:
21
22
23

'
'
'

vec4 color0 = texture2D(u_Sampler0, v_TexCoord);\n' +
vec4 color1 = texture2D(u_Sampler1, v_TexCoord);\n' +
gl_FragColor = color0 * color1;\n' +

There are many possible ways to calculate the final fragment color (gl_FragColor) using
the texels. This sample program uses a component-wise multiplication of both texel colors
because the result is easy to understand. GLSL ES offers a simple way to write this multiplication in a single line as a multiplication of two vec4 variables at line 23 (see Figure 5.36).

*

)

v e c 4 c o lo r 0

r1

g1

b1

a1

v e c 4 c o lo r 1

r2

g2

b2

a2

r1*r2

g1*g2

b1*b2

a1*a2

Figure 5.36

Multiplication of two vec4 variables

Although this sample program uses two texture images, initVertexBuffers() from line 60
is the same as in TexturedQuad.js because it uses the same texture coordinates for both
texture images.
In this sample, initTextures() at line 103 has been modified to repeat the process of
dealing with a texture image twice because now it deals with two images rather than the
single image of the previous example.
Lines 105 and 106 create the two texture objects, one for each texture image. The last
character of each variable name (“0” in texture0 and “1” in texture1) indicates which
texture unit (texture unit 0 or texture unit 1) is used. This naming convention of using
the unit number also applies to the variable names for the storage location of uniform
variables (line 113 and 114) and image objects (lines 120 and 121).
Registration of the event handler (loadTexture()) is the same as in TexturedQuad.js, with
the last argument set to indicate the different texture units:
128
129

image0.onload = function() { loadTexture(gl, n, texture0, u_Sampler0,
➥image0, 0); };
image1.onload = function() { loadTexture(gl, n, texture1,u_Sampler1,
➥image1, 1); };

The request to load the texture images is in lines 131 and 132:

Pasting Multiple Textures to a Shape

187

131
132

image0.src = '../resources/redflower.jpg';
image1.src = '../resources/circle.gif';

In this sample program, the function loadTexture() has to be modified to deal with two
textures. The function is defined from line 138 with its core part as follows:
137 var g_texUnit0 = false, g_texUnit1 = false;
138 function loadTexture(gl, n, texture, u_Sampler, image, texUnit) {
139
gl.pixelStorei(gl.UNPACK_FLIP_Y_WEBGL, 1);// Flip the image's y-axis
140
// Make the texture unit active
141
if (texUnit == 0) {
142
gl.activeTexture(gl.TEXTURE0);
143
g_texUnit0 = true;
144
} else {
145
gl.activeTexture(gl.TEXTURE1);
146
g_texUnit1 = true;
147
}
148
149
gl.bindTexture(gl.TEXTURE_2D, texture); // Bind the texture object
150
151
// Set texture parameters
152
gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_MIN_FILTER, gl.LINEAR);
153
// Set the texture image
154
gl.texImage2D(gl.TEXTURE_2D, 0, gl.RGBA, gl.RGBA, gl.UNSIGNED_BYTE, image);
155
// Set the texture unit number to the sampler
156
gl.uniform1i(u_Sampler, texUnit);
...
164
if (g_texUnit0 && g_texUnit1) {
165
gl.drawArrays(gl.TRIANGLE_STRIP, 0, n); // Draw a rectangle
166
}
167 }

The important difference in loadTexture() is that you cannot predict which texture
image is loaded first because the browser loads them asynchronously. The sample program
handles this by starting to draw only after loading both textures. To do this, it uses two
global variables (g_texUnit0 and g_texUnit1) at line 137 indicating which textures have
been loaded.
These variables are initialized to false at line 137 and changed to true in the if statement at line 141. This if statement checks the variable texUnit passed as the last parameter in loadTexture(). If it is 0, the texture unit 0 is made active and g_texUnit0 is set to
true; if it is 1, the texture unit 1 is made active and then g_texUnit1 is set to true.
Line 156 sets the texture unit number to the uniform variable. Note that the parameter
texUnit of loadTexture() is passed to gl.uniform1i(). After loading two texture images,
the internal state of the system is changed, as shown in Figure 5.37.

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u_S ampler0

JavaS cript
fu n c tio n m a in ( ) {
v a r g l= g e tW e b G L …
…
gl.uniform1i(…);
gl.uniform1i(…);

Figure 5.37

g l.TEXTURE0

g l.TEXTURE1 …

g l .TEXTURE_2D

g l.TEXTURE_2D

F ra g m e n t
S h ad er

.

Texture Object

u_S ampler1

Texture Object

Internal state of WebGL when handling two texture images

Finally, the program invokes the vertex shader after checking whether both texture images
are available at line 165 using g_texUnit0 and g_texUnit1. The images are then combined
as they are mapped to the shape, resulting in the screenshot in Figure 5.34.

Summary
In this chapter, you ventured deep into the WebGL world. At this stage you have acquired
all the basic skills needed to use WebGL to deal with 2D geometric shapes and are ready
for the next step: 3D objects. Fortunately, when you deal with 3D objects instead of 2D
shapes, the way you use shaders is surprisingly similar, so you can quickly apply all the
knowledge you’ve learned so far.
The rest of this book focuses mainly on covering the techniques necessary for managing
3D objects. However, before introducing you to the 3D world, the next chapter will take a
brief tour of the OpenGL ES shading language (GLSL ES), covering some features and functionality that have been only touched on in the chapters so far.

Pasting Multiple Textures to a Shape

189

This page intentionally left blank

Chapter 6

The OpenGL ES Shading Language (GLSL ES)

This chapter takes a break from examining WebGL sample programs and explains the essential
features of the OpenGL ES Shading Language (GLSL ES) in detail.
As you have seen, shaders are the core mechanism within WebGL for constructing 3DCG
programs, and GLSL ES is the dedicated programming language for writing those shader
programs. This chapter covers:

• Data, variables, and variable types
• Vector, matrix, structure, array, and sampler types
• Operators, control flow, and functions
• Attributes, uniform, and varying variables
• Precision qualifiers
• Preprocessor and directives
By the end of this chapter, you will have a good understanding of GLSL ES and how to use it to
write a variety of shaders. This knowledge will help you tackle the more complex 3D manipulations introduced in Chapters 7 through 9. Note that language specifications can be quite dry, and
for some of you, this may be more detail than you need. If so, it’s safe to skip this chapter and
use it as a reference when you look at the examples in the rest of the book.

Recap of Basic Shader Programs
As you can see from Listings 6.1 and 6.2, you can construct shader programs in a similar manner
to constructing programs using the C programming language.

Listing 6.1

Example of a Simple Vertex Shader

// Vertex shader
attribute vec4 a_Position;
attribute vec4 a_Color;
uniform mat4 u_MvpMatrix;
varying vec4 v_Color;
void main() {
gl_Position = u_MvpMatrix * a_Position;
v_Color = a_Color;
}

Variables are declared at the beginning of the code, and then the main() routine defines
the entry point for the program.
Listing 6.2

Example of a Simple Fragment Shader

// Fragment shader
#ifdef GLSL_ES
precision mediump float;
#endif
varying vec4 v_Color;
void main() {
gl_FragColor = v_Color;
}

The version of GLSL ES dealt with in this chapter is 1.00. However, you should note that
WebGL does not support all features defined in GLSL ES 1.001; rather, it supports a subset
of 1.00 with core features needed for WebGL.

Overview of GLSL ES
The GLSL ES programming language was developed from the OpenGL Shading Language
(GLSL) by reducing or simplifying functionality, assuming that the target platforms were
consumer electronics or embedded devices such as smart phones and game consoles.
A prime goal was to allow hardware manufacturers to simplify the hardware needed to
execute GLSL ES programs. This had two key benefits: reducing power consumption by
devices and, perhaps more importantly, reducing manufacturing costs.
GLSL ES supports a limited (and partially extended) version of the C language syntax.
Therefore, if you are familiar with the C language, you’ll find it easy to understand
GLSL ES. Additionally, the shading language is beginning to be used for general-purpose
processing such as image processing and numerical computation (so called GPGPU),
meaning that GLSL ES has an increasingly wide application domain, thus increasing the
benefits of studying the language.
1

http://www.khronos.org/registry/gles/specs/2.0/GLSL_ES_Specification_1.0.17.pdf

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Hello Shader!
By tradition, most programming books begin with a “Hello World!” example, or in our
case the corresponding shader program. However, because you have already seen several
shader programs in previous chapters, let’s skip that and take a look at the basics of GLSL
ES, using Listing 6.1 and Listing 6.2 shown earlier.

Basics
Like many programming languages, you need to pay attention to the following two items
when you write shader programs using GLSL ES:

• The programs are case-sensitive (marina and Marina are different).
• A semicolon (;) must be specified at the end of each command.

Order of Execution
Once a JavaScript program is loaded, program lines are executed in the order in which
they were written—sequentially starting from the first program line. However, like C,
shader programs are executed from the function main() and therefore must have one (and
only one) main() function that cannot have any parameters. Looking back, you can see
that each shader program shown in Listing 6.1 and Listing 6.2 defines a single main().
You must prepend the keyword void to main(), which indicates that the function has
no return value. (See the section “Functions” later in this chapter.) This is different from
JavaScript, where you can define a function using the keyword function, and you don’t
have to worry whether the function returns a value. In GLSL ES, if the function returns a
value, you must specify its data type in front of the function name, or if it doesn’t return
a value, specify void so that the system doesn’t expect a return value.

Comments
As with JavaScript, you can write comments in your shader program, and in fact use the
same syntax as JavaScript. So, the following two types of comment are supported:

• // characters followed by any sequence of characters up to the end of line:
int kp = 496; // kp is a Kaprekar number

• /* characters, followed by any sequence of characters (including new lines), followed
by the */ characters:
/* I have a day off today.
I want to take a day off tomorrow.
*/

Hello Shader!

193

Data (Numerical and Boolean Values)
GLSL ES supports only two data types:

• Numerical value: GLSL ES supports integer numbers (for example, 0, 1, 2) and floating point numbers (for example, 3.14, 29.98, 0.23571). Numbers without a decimal
point (.) are treated as integer numbers, and those with a decimal point are treated
as floating point numbers.

• Boolean value: GLSL ES supports true and false as boolean constants.
GLSL ES does not support character strings, which may initially seem strange but makes
sense for a 3D graphics language.

Variables
As you have seen in the previous chapters, you can use any variable names you want as
long as the name follows the basic naming rules:

• The character set for variables names contains only the letters a–z, A–Z, the underscore (_), and the numbers 0–9.

• Numbers are not allowed to be used as the first character of variable names.
• The keywords shown in Table 6.1 and the reserved keywords shown in Table 6.2 are
not allowed to be used as variable names. However, you can use them as part of the
variable name, so the variable name if will result in error, but iffy will not.

• Variable names starting with gl_, webgl_, or _webgl_ are reserved for use by OpenGL
ES. No user-defined variable names may begin with them.

Table 6.1

Keywords Used in GLSL ES

attribute

bool

break

bvec2

bvec3

bvec4

const

continue

discard

do

else

false

float

for

highp

if

in

inout

Int

invariant

ivec2

ivec3

ivec4

lowp

mat2

mat3

mat4

medium

out

precision

return

sampler2D

samplerCube

struct

true

uniform

varying

vec2

vec3

vec4

void

while

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The OpenGL ES Shading Language (GLSL ES)

Table 6.2

Reserved Keywords for Future Version of GLSL ES

asm

cast

class

default

double

dvec2

dvec3

dvec4

enum

extern

external

fixed

flat

fvec2

fvec3

fvec4

goto

half

hvec2

hvec3

hvec4

inline

input

interface

long

namespace

noinline

output

packed

public

sampler1D

sampler1DShadow

sampler2DRect

sampler2DRectShadow

sampler2DShadow

sampler3D

sampler3DRect

short

sizeof

static

superp

switch

template

this

typedef

union

unsigned

using

volatile

GLSL ES Is a Type Sensitive Language
GLSL ES does not require the use of var to declare variables, but it does require you to
specify the type of data a variable will contain. As you have seen in the sample programs,
you declare variables using the form
 

such as vec4 a_Position.
As discussed, when you define a function like main(), you must also specify the data type
of the return value of the function. Equally, the type of data on the left side of the assignment operation (=) and that of data on the right side must have the same type; otherwise,
it will result in an error.
For these reasons, GLSL ES is called a type sensitive language, meaning that it belongs to
a class of languages that require you to specify and pay attention to types.

Basic Types
GLSL ES supports the basic data types shown in Table 6.3.

Basic Types

195

Table 6.3

GLSL Basic Types

Type

Description

float

The data type for a single floating point number. It indicates the variable will contain
a single floating point number.

int

The data type for a single integer number. It indicates the variable will contain a
single integer number.

bool

The data type for a boolean value. It indicates the variable will contain a boolean
value.

Specifying the data type for variables allows the WebGL system to check errors in advance
and process the program efficiently. The following are examples of variable declarations
using basic types.
float klimt;
int utrillo;
bool doga;

// The variable will contain a single floating number
// The variable will contain a single integer number
// The variable will contain a single boolean value

Assignment and Type Conversion
Assignments of values to variables are performed using the assignment operator (=). As
mentioned, because GLSL ES is a type-sensitive language, if the data type of the left-side
variable is not equal to that of the assigned data (or variable), it will result in an error:
int i
float
float
float

= 8;
f1 = 8;
f2 = 8.0;
f3 = 8.0f;

//
//
//
//

OK
Error
OK
Error: Expressions like 8.0f used in C are not allowed.

Semantically, 8 and 8.0 are the same values. However, when you assign 8 to a floating
point variable f1, it will result in an error. In this case, you would see the following error
message:
failed to compile shader: ERROR: 0:11: '=' : cannot convert from 'const mediump int'
to 'float'.

If you want to assign an integer number to a floating point variable, you need to convert
the integer number to a floating point number. This conversion is called type conversion.
To convert an integer into a floating point number, you can use the built-in function
float(), as follows:
int i = 8;
float f1 = float(i); // 8 is converted to 8.0 and assigned to f1
float f2 = float(8); // equivalent to the above operation

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The OpenGL ES Shading Language (GLSL ES)

GLSL ES supports a number of other built-in functions for type conversion, which are
shown in Table 6.4.
Table 6.4

The Built-In Functions for Type Conversion

Conversion

Function

Description

To an integer
number

int(float)

The fractional part of the floating-point value is
dropped (for example, 3.14 → 3).

int(bool)

true is converted to 1, or false is converted to 0.

float(int)

The integer number is converted to a floating point
number (for example, 8 → 8.0).

To a floating
point number

true is converted to 1.0, or false is converted to

float(bool)

0.0.
To a boolean
value

bool(int)

0 is converted to false, or non-zero values are
converted to true.

bool(float)

0.0 is converted to false, or non-zero values are
converted to true.

Operations
The operators applicable to the basic types are similar to those in JavaScript and are shown
in Table 6.5.
Table 6.5

The Operators Available for the Basic Types

Operator

Operation

Applicable Data Type

-

Negation (for example,
for specifying a negative
number)

int or float.

*

Multiplication

/

Division

int or float. The data type of the
result of the operation is the same
as operands.

+

Addition

-

Subtraction

++

Increment (postfix and
prefix)

--

Decrement (postfix and
prefix)

=

Assignment

int, float, or bool

+= -= *= /=

Arithmetic assignment

int or float.

int or float. The data type of the
result of the operation is the same
as operands.

Basic Types

197

Operator

Operation

Applicable Data Type

< > <= >=

Comparison

int or float.

== !=

Comparison (equality)

int, float, or bool.

!

Not

&&

Logical and

bool or an expression that results
in bool [1].

||

Logical inclusive or

^^

Logical exclusive or [2]

condition?
expression1:expression2

Ternary selection

condition is bool or an expression
that results in bool. Data types
other than array can be used in
expression1 and expression2.

[1] The second operand in a logical and (&&) operation is evaluated if and only if the first operand
evaluates to true. The second operand in a logical or (||) operation is evaluated if and only if the
first operand evaluates to false.
[2] If either the left-side condition or the right-side one is true, the result is true. If both sides are
true, the result is false.

The followings are examples of basic operations:
int i1 = 954, i2 = 459;
int kp = i1 - i2; // 495 is assigned to kp.
float f = float(kp) + 5.5; // 500.5 is assigned to f.

Vector Types and Matrix Types
GLSL ES supports vector and matrix data types which, as you have seen, are useful when
dealing with computer graphics. Both these types contain multiple data elements. A
vector type, which arranges data in a list, is useful for representing vertex coordinates or
color data. A matrix arranges data in an array and is useful for representing transformation
matrices. Figure 6.1 shows an example of both types.

(3

7 1)

Figure 6.1

3 7 1
1 5 3
4 0 7
A vector and a matrix

GLSL ES supports a variety of vector or matrix types, as shown in Table 6.6.

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The OpenGL ES Shading Language (GLSL ES)

Table 6.6

Vector Types and Matrix Types

Category

Types in GLSL ES

Description

Vector

vec2, vec3, vec4

The data types for 2, 3, and 4 component vectors of
floating point numbers

ivec2, ivec3, ivec4

The data types for 2, 3, and 4 component vectors of
integer numbers

bvec2, bvec3, bvec4

The data types for 2, 3, and 4 component vectors of
boolean values

mat2, mat3, mat4

The data type for 2×2, 3×3, and 4×4 matrix of floating
point numbers (with 4, 9, and 16 elements, respectively)

Matrix

The following examples show the use of the vector and matrix types:
vec3 position;

//
//
ivec2 offset;
//
//
mat4 mvpMatrix; //

variable for
For example:
variable for
For example:
the variable

3-component vector of float
(10.0, 20.0, 30.0)
2-component vector of integer
(10, 20)
for 4×4 matrix of float

Assignments and Constructors
Assignment of data to variables of the type vector or matrix is performed using the =
operator. Remember that the type of data on the left side of the assignment operation and
that of the data/variable on the right side must be the same. In addition, the number of
elements on the left side of the assignment operation must be equal to that of the data/
variable on the right side. To illustrate that, the following example will result in an error:
vec4 position = 1.0; // vec4 variable requires four floating point numbers

In this case, because a vec4 variable requires four floating point numbers, you need to pass
four floating numbers in some way. A solution is to use the built-in functions with the
same name of the data type so; for example, in the case of vec4, you can use the constructor vec4(). (See Chapter 2, “Your First Step with WebGL.”) For example, to assign 1.0, 2.0,
3.0, and 4.0 to a variable of type vec4, you can use vec4() to bundle them into a single
data element as follows:
vec4 position = vec4(1.0, 2.0, 3.0, 4.0);

Functions for making a value of the specified data type are called constructor functions,
and the name of the constructor is always identical to that of the data type.

Vector Types and Matrix Types

199

Vector Constructors
Vectors are critical in GLSL ES so, as you’d imagine, there are multiple ways to specify
arguments to a vector constructor. For example:
vec3 v3 = vec3(1.0, 0.0, 0.5); // sets v3 to(1.0, 0.0, 0.5)
vec2 v2 = vec2(v3); // sets v2 to (1.0, 0.0) using the 1st and 2nd elements of v3
vec4 v4 = vec4(1.0); // sets v4 to (1.0, 1.0, 1.0, 1.0)

In the second example, the constructor ignores the third element of v3, and only the
first and second elements of v3 are used to create the new vector. Similarly, in the third
example, if a single value is specified to a vector constructor, the value is used to initialize
all components of the constructed vector. However, if more than one value is specified
to a vector constructor but the number of the values is less than the number of elements
required by the constructor, it will result in an error.
Finally, a vector can be constructed from multiple vectors:
vec4 v4b = vec4(v2, v4);

// sets (1.0, 0.0, 1.0, 1.0) to v4b

The rule here is that the vector is filled with values from the first vector (v2), and then
any missing values are supplied by the second vector (v4).
Matrix Constructors
Constructors are also available for matrices and operate in a similar manner to vector
constructors. However, you should make sure the order of elements stored in a matrix is
in a column major order. (See Figure 3.27 for more details of “column-major order.”) The
following examples show different ways of using the matrix constructor:

• If multiple values are specified to a matrix constructor, a matrix is constructed using
them in column major order:
mat4 m4 = mat4 (

1.0, 2.0,
5.0, 6.0,
9.0, 10.0,
13.0, 14.0,

3.0,
7.0,
11.0,
15.0,

4.0,
8.0,
12.0,
16.0 );

⎡
⎢
⎢
⎢
⎢
⎣

1.0
2.0
3.0
4.0

5.0 9.0 13.0 ⎤
⎥
6.0 10.0 14.0 ⎥
7.0 11.0 15.0 ⎥
8.0 12.0 16.0 ⎥⎦

• If multiple vectors are specified to a matrix constructor, a matrix is constructed using
the elements of each vector in column major order:
// two vec2
vec2 v2_1 =
vec2 v2_2 =
mat2 m2_1 =

are used to construct a mat2
vec2(1.0, 3.0);
vec2(2.0, 4.0);
mat2(v2_1, v2_2); // 1.0 2.0
// 3.0 4.0
// vec4 is used to construct mat2
vec4 v4 = vec4(1.0, 3.0, 2.0, 4.0);

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The OpenGL ES Shading Language (GLSL ES)

mat2 m2_2 = mat2(v4);

// 1.0 2.0
// 3.4 4.0

• If multiple values and multiple vectors are specified to a matrix constructor, a matrix
is constructed using them in column major order:
// Two floating point numbers and vec2 are used to construct a mat2
mat2 m2 = mat2(1.0, 3.0, v2_2); // 1.0 2.0
// 3.0 4.0

• If a single value is specified to a matrix constructor, a matrix is constructed using the
value as its diagonal elements:
mat4 m4 = mat4(1.0);

//
//
//
//

1.0
0.0
0.0
0.0

0.0
1.0
0.0
0.0

0.0
0.0
1.0
0.0

0.0
0.0
0.0
1.0

Similar to a vector constructor, if an insufficient number of values is specified to the
constructor (but more than one), it will result in an error.
mat4 m4 = mat4(1.0, 2.0, 3.0); // Error. mat4 requires 16 elements.

Access to Components
To access the components in a vector or matrix, you can use the operators . and [], as
shown in the following subsections.
The . Operator
An individual component in a vector can be accessed by the variable name followed by
period (.) and then the component name, as shown in Table 6.7.
Table 6.7

Component Names

Category

Description

x, y, z, w

Useful for accessing vertex coordinates.

r, g, b, a

Useful for accessing colors.

s, t, p, q

Useful for accessing texture coordinates. (Note that this book uses only s and
t. p is used instead of r because r is used for colors.)

Because vectors are used for storing various types of data such as vertex coordinates,
colors, and texture coordinates, three types of component names are supported to increase

Vector Types and Matrix Types

201

the readability of programs. However, any of the component names x, r, or s accesses the
first component; any of y, g, or t accesses the second one; and so on, so you can use them
interchangeably if you prefer. For example:
vec3 v3 = vec3(1.0, 2.0, 3.0);
float f;

// sets v3 to(1.0, 2.0, 3.0)

f = v3.x; // sets f to 1.0
f = v3.y; // sets f to 2.0
f = v3.z; // sets f to 3.0
f = v3.r; // sets f to 1.0
f = v3.s; // sets f to 1.0

As you can see from the comments of these examples, x, r, and s have different names
but always access the first component. Attempting to access a component beyond the
number of components in the vector will result in an error:
f = v3.w; // w requires access to the fourth element, which doesn't exist.

Multiple components can be selected by appending their names (from the same name set)
after the period (.). This is known as swizzling. In the following example, x, y, z, and w
will be used, but other sets of component names have the same effect:
vec2
v2 =
v2 =
v2 =
v2 =
v2 =

v2;
v3.xy;
v3.yz;
v3.xz;
v3.yx;
v3.xx;

//
//
//
//
//

sets
sets
sets
sets
sets

v2
v2
v2
v2
v2

to
to
to
to
to

(1.0,
(2.0,
(1.0,
(2.0,
(1.0,

2.0)
3.0).
3.0).
1.0).
1.0).

Any
You
You
You

component can be omitted
can skip any component.
can reverse the order.
can repeat any component.

vec3 v3a;
v3a = v3.zyx; // sets v3a to (3.0, 2.0, 1.0). You can use all names.

The component name can also be used in the left-side expression of an assignment operator (=):
vec4 position = vec4(1.0, 2.0, 3.0, 4.0);
position.xw = vec2(5.0, 6.0); // position = (5.0, 2.0, 3.0, 6.0)

Remember, the component names must come from the same set so, for example, v3.was is
not allowed.

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The [ ] Operator
In addition to the . operator, the components of a vector or a matrix can be accessed
using the array indexing operator []. Note that the elements in a matrix are also read out
in column major order. Just like JavaScript, the index starts from 0, so applying [0] to a
matrix selects the first column in the matrix, [1] selects the second one, [2] selects the
third one, and so on. The following shows an example:
mat4 m4 = mat4 ( 1.0, 2.0, 3.0, 4.0,
5.0, 6.0, 7.0, 8.0,
9.0, 10.0, 11.0, 12.0,
13.0, 14.0, 15.0, 16.0);
vec4 v4 = m4[0]; // Retrieve the 1st column from m4: (1.0, 2.0, 3.0, 4.0)

In addition, two [] operators can be used to select a column and then a row of a matrix:
float m23 = m4[1][2]; // sets m23 to the third component of the second
// column of m4 (7.0).

A component name can be used to select a component in conjunction with the [] operator, as follows:
float m32 = m4[2].y; // sets m32 to the second component of the third
// column of m4 (10.0).

One restriction is that only a constant index can be specified as the index number in the
[] operator. The constant index is defined as

• A integral literal value (for example, 0 or 1)
• A global or local variable qualified as const, excluding function parameters (see the
section “const Variables”)

• Loop indices (see the section “Conditional Control Flow and Iteration”)
• Expressions composed from any of the preceding
The following examples use the type int constant index:
const int index = 0;

// "const" keyword specifies the variable is a
// read-only variable.
vec4 v4a = m4[index]; // is the same as m4[0]

The following example uses an expression composed of constant indices as an index.
vec4 v4b = m4[index + 1]; // is the same as m4[1]

Vector Types and Matrix Types

203

Remember, you cannot use an int variable without the const qualifier as an index
because it is not a constant index (unless it is a loop index):
int index2 = 0;
vec4 v4c = m4[index2]; // Error: because index2 is not a constant index.

Operations
You can apply the operators shown in Table 6.8 to a vector or a matrix. These operators
are similar to the operators for basic types. Note that the only comparative operators available for a vector and matrix are == and !=. The <, >, <=, and >= operators cannot be used
for comparisons of vectors or matrices. In such cases, you can use built-in functions such
as lessThan(). (See Appendix B, “Built-In Functions of GLSL ES 1.0.”)
Table 6.8

The Operators Available for a Vector and a Matrix

Operators

Operation

Applicable Data Types

*

Multiplication

/

Division

vec[234] and mat[234]. The operations on vec[234] and mat[234] will be

+

Addition

-

Subtraction

++

Increment (postfix and prefix)

vec[234] and mat[234]. The data type of

--

Decrement (postfix and prefix)

the result of this operation is the same as
the operands.

=

Assignment

vec[234] and mat[234].

+=, -=,
*=, /=

Arithmetic assignment

vec[234] and mat[234].

==, !=

Comparison

vec[234] and mat[234]. With ==, if all
components of the operands are equal,
the result is true. For !=, if any of components of the operands are not equal, then
the result is true [1].

explained below.
The data type of the result of operation is
the same as the operands.

[1] If you want component-wise equality, you can use the built-in function equal() or
notEqual(). (See Appendix B.)

Note that when an arithmetic operator operates on a vector or a matrix, it is operating
independently on each component of the vector or matrix in component-wise order.
Examples
The following examples show frequently used cases. In the examples, we assume that the
types of variables are defined as follows:

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vec3 v3a, v3b, v3c;
mat3 m3a, m3b, m3c;
float f;

Operations on a Vector and Floating Point Number
An example showing the use of the + operator:
// The following example uses the + operator, but the -, *, and /
// operators also have the same effect.
v3b = v3a + f;
// v3b.x = v3a.x + f;
// v3b.y = v3a.y + f;
// v3b.z = v3a.z + f;

For example, v3a = vec3(1.0, 2.0, 3.0) and f = 1.0 will result in v3b=(2.0, 3.0,
4.0).

Operations on Vectors
These operators operate on each component of a vector:
// The following example uses the + operator, but the -, *, and /
// operators also have the same effect.
v3c = v3a + v3b; // v3a.x + v3b.x;
// v3a.y + v3b.y;
// v3a.z + v3b.z;

For example, v3a = vec3(1.0, 2.0, 3.0) and v3b = vec3(4.0, 5.0, 6.0) will result in
v3c=(5.0, 7.0, 9.0).
Operations on a Matrix and a Floating Point Number
These operators operate on each component of the matrix:
// The following example uses the + operator, but the
// operators also have the same effect.
m3b = m3a * f;
// m3b[0].x = m3a[0].x * f; m3b[0].y
// m3b[0].z = m3a[0].z * f;
// m3b[1].x = m3a[1].x * f; m3b[1].y
// m3b[1].z = m3a[1].z * f;
// m3b[2].x = m3a[2].x * f; m3b[2].y
// m3b[2].z = m3a[2].z * f;

-, *, and /
= m3a[0].y * f;
= m3a[1].y * f;
= m3a[2].y * f;

Multiplication of a Matrix and a Vector
For multiplication, the result is the sum of products of each element in a matrix and
vector. This result is the same as Equation 3.5 that you saw back in Chapter 3, “Drawing
and Transforming Triangles”:

Vector Types and Matrix Types

205

v3b = m3a * v3a;

// v3b.x = m3a[0].x * v3a.x + m3a[1].x *
//
+ m3a[2].x *
// v3b.y = m3a[0].y * v3a.x + m3a[1].y *
//
+ m3a[2].y *
// v3b.z = m3a[0].z * v3a.x + m3a[1].z *
//
+ m3a[2].z *

v3a.y
v3a.z;
v3a.y
v3a.z;
v3a.y
v3a.z;

Multiplication of a Vector and a Matrix
Multiplication of a vector and a matrix is possible, as you can see from the following
expressions. Note that this result is different from that when multiplying a matrix by a
vector:
v3b = v3a * m3a; // v3b.x = v3a.x * m3a[0].x + v3a.y *
//
+ v3a.z *
// v3b.y = v3a.x * m3a[1].x + v3a.y *
//
+ v3a.z *
// v3b.z = v3a.x * m3a[2].x + v3a.y *
//
+ v3a.z *

m3a[0].y
m3a[0].z;
m3a[1].y
m3a[1].z;
m3a[2].y
m3a[2].z;

Multiplication of Matrices
This is the same as Equation 4.4 in Chapter 4, “More Transformations and Basic
Animation”:
m3c = m3a * m3b; // m3c[0].x = m3a[0].x * m3b[0].x + m3a[1].x *
//
+ m3a[2].x *
// m3c[1].x = m3a[0].x * m3b[1].x + m3a[1].x *
//
+ m3a[2].x *
// m3c[2].x = m3a[0].x * m3b[2].x + m3a[1].x *
//
+ m3a[2].x *

m3b[0].y
m3b[0].z;
m3b[1].y
m3b[1].z;
m3b[2].y
m3b[2].z;

// m3c[0].y = m3a[0].y * m3b[0].x + m3a[1].y *
//
+ m3a[2].y *
// m3c[1].y = m3a[0].y * m3b[1].x + m3a[1].y *
//
+ m3a[2].y *
// m3c[2].y = m3a[0].y * m3b[2].x + m3a[1].y *
//
+ m3a[2].y *

m3b[0].y
m3b[0].z;
m3b[1].y
m3b[1].z;
m3b[2].y
m3b[2].z;

// m3c[0].z = m3a[0].z * m3b[0].x + m3a[1].z *
//
+ m3a[2].z *
// m3c[1].z = m3a[0].z * m3b[1].x + m3a[1].z *
//
+ m3a[2].z *
// m3c[2].z = m3a[0].z * m3b[2].x + m3a[1].z *
//
+ m3a[2].z *

m3b[0].y
m3b[0].z;
m3b[1].y
m3b[1].z;
m3b[2].y
m3b[2].z;

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Structures
GLSL ES also supports user-defined types, called structures, which aggregate other already
defined types using the keyword struct. For example:
struct light {
// defines the structure "light"
vec4 color;
vec3 position;
}
light l1, l2;
// declares variable "l1" and "l2" of the type "light"

This example defines the new structure type light that consists of two members: the variable color and position. Then two variables l1 and l2 of type light are declared after the
definition. Unlike C, the typedef keyword is not necessary because, by default, the name
of the structure becomes the name of the type.
In addition, as a convenience, variables of the new type can be declared with the definition of the structure, as follows:
struct light {
vec4 color;
vec3 position;
} l1;

//
//
//
//

declares
color of
position
variable

structure and its variable all together
a light
of a light
"l1" of the structure

Assignments and Constructors
Structures support the standard constructor, which has the same name as the structure.
The arguments to the constructor must be in the same order and of the same type as they
were declared in the structure. Figure 6.2 shows an example.
l1 = l igh t (ve c4 (0.0, 1.0, 0.0, 1.0),ve c3(8.0, 3.0, 0.0));
c o lo r
p o s itio n

Figure 6.2

A constructor of structure

Access to Members
Each member of a structure can be accessed by appending the variable name with a period
(.) and then the member name. For example:
vec4 color = l1.color;
vec3 position = l1.position;

Structures

207

Operations
For each member in the structure, you can use any operators allowed for that member’s
type. However, the operators allowed for the structure itself are only the assignment (=)
and comparative operators (== and !=); see Table 6.9.
Table 6.9

The Operators Available for a Structure

Operator

Operation

Description

=

Assignment

==,!=

Comparison

The assignment and comparison operators are not allowed for
the structures that contain arrays or sampler types

When using the == operator, the result is true if, and only if, all the members are
component-wise equal. When using the !=, the result is false if one of the members is
not component-wise equal.

Arrays
GLSL ES arrays have a similar form to the array in JavaScript, with only one-dimensional
arrays being supported. Unlike arrays in JavaScript, the new operator is not necessary to
create arrays, and methods such as push() and pop() are not supported. The arrays can be
declared by a name followed by brackets ([]) enclosing their sizes. For example:
float floatArray[4]; // declares an array consisting of four floats
vec4 vec4Array[2];
// declares an array consisting of two vec4s

The array size must be specified as an integral constant expression greater than zero
where the integral constant expression is defined as follows:

• A numerical value (for example, 0 or 1)
• A global or local variable qualified as const, excluding function parameters (see the
section “const Variables”)

• Expressions composed of both of the above
Therefore, the following will result in an error:
int size = 4;
vec4 vec4Array[size]; // Error. If you declare "const int size = 4;"
// it will not result in an error

Note that arrays cannot be qualified as const.

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Array elements can be accessed using the array indexing operator ([]). Note that, like C,
the index starts from 0. For example, the third element of the float Array defined earlier
can be accessed as follows:
float f = floatArray[2];

Only an integral constant expression or uniform variable (see the section “Uniform
Variables”) can be used as an index of an array. In addition, unlike JavaScript or C, an
array cannot be initialized at declaration time. So each element of the array must be
initialized explicitly as follows:
vec4Array[0] = vec4(4.0, 3.0, 6.0, 1.0);
vec4Array[1] = vec4(3.0, 2.0, 0.0, 1.0);

Arrays support only [] operators. However, elements in an array do support the standard
operators available for their type. For example, the following operator can be applied to
the elements of floatArray or vec4Array:
// multiplies the second element of floatArray by 3.14
float f = floatArray[1] * 3.14;
// multiplies the first element of vec4Array by vec4(1.0, 2.0, 3.0, 4.0);
vec4 v4 = vec4Array[0] * vec4(1.0, 2.0, 3.0, 4.0);

Samplers
GLSL ES supports a dedicated type called sampler for accessing textures. (See Chapter 5,
“Using Colors and Texture Images.”) Two types of samplers are available: sampler2D and
samplerCube. Variables of the sampler type can be used only as a uniform variable (see
the section “Uniform Variables”) or an argument of the functions that can access textures
such as texture2D(). (See Appendix B.) For example:
uniform sampler2D u_Sampler;

In addition, the only value that can be assigned to the variable is a texture unit number,
and you must use the WebGL method gl.uniform1i() to set the value. For example,
TexturedQuad.js in Chapter 5 uses gl.uniform1i(u_Sampler, 0) to pass the texture unit
0 to the shader.
Variables of type sampler are not allowed to be operands in any expressions other than =,
==, and !=.
Unlike other types explained in the previous sections, the number of sampler type variables is limited depending on the shader type (see Table 6.10). In the table, the keyword
mediump is a precision qualifier. (This qualifier is explained in detail in the section
“Precision Qualifiers,” toward the end of this chapter.)

Samplers

209

Table 6.10

Minimum Number of Variables of the Sampler Type

Shaders that Use the Built-In Constants Representing the
Variable
Maximum Number

Minimum Number

Vertex shader

const mediump int
gl_MaxVertexTextureImageUnits

0

Fragment shader

const mediump int
gl_MaxTextureImageUnits

8

Precedence of Operators
Operator precedence is shown in Table 6.11. Note the table contains several operators that
are not explained in this book but are included for reference.
Table 6.11

The Precedence of Operators

Precedence

Operators

1

parenthetical grouping (())

2

function calls, constructors (()), array indexing ([]), period (.)

3

increment/decrement (++, --), negate (-), inverse(~), not(!)

4

multiplication (*), division (/), remainder (%)

5

addition (+), subtraction (-)

6

bit-wise shift (<<, >>)

7

comparative operators (<, <=, >=, >)

8

equality (==, !=)

9

bit-wise and (&)

10

bit-wise exclusive or (^)

11

bit-wise or (|)

12

and (&&)

13

exclusive or (^^)

14

or (||)

15

ternary selection (? :)

16

assignment (=), arithmetic assignments (+=, -=, *=, /=, %=,
<<=, >>=, &=, ^=, |=)

17

sequence(,)

Bold font indicates operators reserved for future versions of GLSL.

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Conditional Control Flow and Iteration
Conditional control flow and iteration in the shading language are almost the same as in
JavaScript or C.

if Statement and if-else Statement
A conditional control flow can use either if or if-else. An if-else statement follows the
pattern shown here:
if (conditional-expression1) {
commands here are executed if conditional-expression1 is true.
} else if (conditional-expression2) {
commands here are executed if conditional-expression1 is false but conditionalexpression2 is true.
} else {
commands here are executed if conditional-expression1 is false and conditionalexpression2 is false.
}

The following shows a code example using the if-else statement:
if(distance < 0.5) {
gl_FragColor = vec4(1.0, 0.0, 0.0, 1.0); // red
} else {
gl_FragColor = vec4(0.0, 1.0, 0.0, 1.0); // green
}

As shown in this example, the conditional expression in the if or if-else statement must
be either a boolean value or an expression that becomes a boolean value. Boolean vector
types, such as bvec2, are not allowed in the conditional expression.
Switch statement are not allowed, and you should note that usage of the if or if-else
statement will slow down the shaders.

for Statement
The for statement can be used as follows:
for (for-init-statement; conditional-expression; loop-index-expression) {
the commands which you want to execute repeatedly.
}

For example:
for (int i = 0; i < 3; i++) {
sum += i;
}

Conditional Control Flow and Iteration

211

Note that the loop index (i in the preceding example) of the for statement can be
declared only in the for-init-statement. The conditional-expression can be omitted, and an
empty condition becomes true. The for statement has the following restrictions:

• Only a single loop index is allowed. The loop index must have the type int or
float.

• loop-index-expression must have one of the following forms (supposing that i is a
loop index):
i++, i--, i+=constant-expression, i-=constant-expression

• conditional-expression is a comparison between a loop index and an integral constant
expression. (See the section “Array.”)

• Within the body of the loop, the loop index cannot be assigned.
These limitations are in place so that the compiler can perform inline expansion of for
statements.

continue, break, discard Statements
Just like JavaScript or C, continue and break statements are allowed only within a for
statement and are generally used within if statements:
continue skips the remainder of the body of the innermost loop containing the
continue, increases/decreases the loop index, and then moves to the next loop.
break exits the innermost loop containing the break. No further execution of the loop is

done.
The following show examples of the continue statement:
for (int i = 0; i < 10; i++) {
if (i == 8) {
continue; // skips the remainder of the body of the innermost loop
}
// When i == 8, this line is not executed
}

The following shows an example of the break statement:
for (int i = 0; i < 10; i++) {
if (i == 8) {
break; // exits "for" loop
}
// When i >= 8, this line is not executed.
}
// When i == 8, this line is executed.

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The discard statement is only allowed in fragment shaders and discards the current fragment, abandoning the operation on the current fragment and skipping to the next fragment. The use of discard will be explained in more detail in the section “Make a Rounded
Point” in Chapter 10, “Advanced Techniques.”

Functions
In contrast to the way functions are defined in JavaScript, the functions in GLSL ES are
defined in the same manner as in C. For example:
returnType functionName(type0 arg0, type1 arg1, ..., typen argn) {
do some computation
return returnValue;
}

Argument types must use one of the data types explained in this chapter, and like main(),
functions with no arguments are allowed. When the function returns no value, the return
statement does not need to be included. In this case, returnType must be void. You can
also specify a structure as the returnValue, but the structure returned cannot contain an
array.
The following example shows a function to convert an RGBA value into a luminance
value:
float luma (vec4 color) {
float r = color.r;
float g = color.g;
float b = color.b;
return 0.2126 * r + 0.7162 * g + 0.0722 * b;
// The preceding four lines could be rewritten as follows:
// return 0.2126 * color.r + 0.7162 * color.g + 0.0722 * color.b;
}

You can call the function declared above in the same manner as in JavaScript or C by
using its name followed by a list of arguments in parentheses:
attribute vec4 a_Color; // (r, g, b, a) is passed
void main() {
...
float brightness = luma(a_Color);
...
}

Note that an error will result if, when called, argument types do not match the declared
parameter types. For example, the following will result in an error because the type of the
parameter is float, but the caller passes an integer:

Functions

213

float square(float value) {
return value * value;
}
void main() {
...
float x2 = square(10); // Error: Because 10 is integer. 10.0 is OK.
...
}

As you can see from the previous examples, functions work just like those in JavaScript
or C except that you cannot call the function itself from inside the body of the function
(that is, a recursive call of the function isn’t allowed). For the more technically minded,
this is because the compilers can in-line function calls.

Prototype Declarations
When a function is called before it is defined, it must be declared with a prototype. The
prototype declaration tells WebGL in advance about the types of parameters and the
return value of the function. Note that this is different from JavaScript, which doesn’t
require a prototype. The following is an example of a prototype declaration for luma(),
which you saw in the previous section:
float luma(vec4); // a prototype declaration
main() {
...
float brightness = luma(color); // luma() is called before it is defined.
...
}
float luma (vec4 color) {
return 0.2126 * color.r + 0.7162 * color.g + 0.0722 * color.b;
}

Parameter Qualifiers
GLSL ES supports qualifiers for parameters that control the roles of parameters within a
function. They can define that a parameter (1) is to be passed into a function, (2) is to be
passed back out of a function, and (3) is to be passed both into and out of a function. (2)
and (3) can be used just like a pointer in C. These are shown in Table 6.12.

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Table 6.12

Parameter Qualifiers

Qualifiers

Roles

Description

in

Passes a value into the function

The parameter is passed by value.
Its value can be referred to and
modified in the function. The caller
cannot refer to the modification.

const in

Passes a value into the function

The parameter is passed by
constant value. Its value can be
referred to but cannot be modified.

out

Passes a value out of the function

The parameter is passed by reference. If its value is modified, the
caller can refer to the modification.

inout

Passes a value both into/out of the
function

The parameter is passed by reference, and its value is copied in the
function. Its value can be referred
to and modified in the function. The
caller can also refer to the modification.



Passes a value into the function

Same as in.

For example, luma() can return the result of its calculation using a parameter qualified by
out instead of a return value, as follows:
void luma2 (in vec3 color, out float brightness) {
brightness = 0.2126 * color.r + 0.7162 * color.g + 0.0722 * color.b;
}

Because the function itself no longer returns a value, the return type of this function is
changed from float to void. Additionally, the qualifier in, in front of the first parameter,
can be omitted because in is a default parameter qualifier.
This function can be used as follows:
luma2(color, brightness); // the result is stored into "brightness"
// same as brightness = luma(color)

Built-In Functions
In addition to user-defined functions, GLSL ES supports a number of built-in functions
that perform operations frequently used in computer graphics. Table 6.13 gives an overview of the built-in functions in GLSL ES, and you can look at Appendix B for the detailed
definition of each function.

Built-In Functions

215

Table 6.13

Built-In Functions in GLSL ES

Category

Built-In Functions

Angle functions

radians (converts degrees to radians), degrees (converts radians to

degrees)
Trigonometry
functions

sin (sine function), cos (cosine function), tan (tangent function), asin
(arc sine function), acos (arc cosine function), and atan (arc tangent
function)

Exponential
functions

pow (xy), exp (natural exponentiation), log (natural logarithm), exp2 (2x),
log2 (base 2 logarithm), sqrt (square root), and inversesqrt (inverse
of sqrt)

Common functions

abs (absolute value), min (minimum value), max (maximum value), mod
(remainder), sign (sign of a value), floor (floor function), ceil (ceil
function), clamp (clamping of a value), mix (linear interpolation), step
(step function), smoothstep (Hermite interpolation), and fract (frac-

tional part of the argument)
Geometric functions

length (length of a vector), distance (distance between two points),
dot (inner product), cross (outer product), normalize (vector with
length of 1), reflect (reflection vector), and faceforward (converting

normal when needed to “faceforward”)
Matrix functions

matrixCompMult (component-wise multiplication)

Vector relational
functions

lessThan (component-wise “<”), lessThanEqual (component-wise
“<=”), greaterThan (component-wise “>”), greaterThanEqual
(component-wise “>=”), equal (component-wise “==”), notEqual
(component-wise “!=”), any (true if any component is true), all (true if
all components are true), and not (component-wise logical complement)

Texture lookup
functions

texture2D (texture lookup in the 2D texture), textureCube (texture
lookup in the cube map texture), texture2DProj (projective version of
texture2D()), texture2DLod (level of detail version of texture2D()),
textureCubeLod (lod version of textureCube()), and
texture2DProjLod (projective version of texture2DLod())

Global Variables and Local Variables
Just like JavaScript or C, GLSL ES supports both global variables and local variables. Global
variables can be accessed from anywhere in the program, and local variables can be
accessed only from within a limited portion of the program.
In GLSL ES, in a similar manner to JavaScript or C, variables declared “outside” a function
become global variables, and variables declared “inside” a function become local variables.
The local variables can be accessed only from within the function containing them. For
this reason, the attribute, uniform, and varying variables described in the next section
must be declared as global variables because they are accessed from outside the function.
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Storage Qualifiers
As explained in the previous chapters, GLSL ES supports storage qualifiers for attribute,
uniform, and varying variables (see Figure 6.3). In addition, a const qualifier is supported
to specify a constant variable to be used in a shader program.
JavaScript
Color Buffer

Vertex
Shader

varying variable

Figure 6.3

unifo rm var iabl e

Fragment
Shader

gl_FragColor

function main() {
var gl=getWebGL…
…
i n itShaders ( … ) ;
…
}

u n if o r m va ri ab l e

gl_Position

a ttrib ute var ia ble

varying variable

Attribute, uniform, and varying variables

const Variables
Unlike JavaScript, GLSL ES supports the const qualifier to specify a constant variable, or
one whose value cannot be modified.
The const qualifier is specified in front of the variable type, just like an attribute variable. Variables qualified by const must be initialized at their declaration time; otherwise,
they are unusable because no data can be assigned to them after their declaration. Some
examples include:
const int lightspeed = 299792458;
// light speed (m/s)
const vec4 red = vec4(1.0, 0.0, 0.0, 1.0); // red
const mat4 identity = mat4(1.0);
// identity matrix

Assigning data to the variable qualified by const will result in an error. For example:
const int lightspeed;
lightspeed = 299792458;

will result in the following error message:
failed to compile shader: ERROR: 0:11: 'lightspeed' : variables
with qualifier 'const' must be initialized
ERROR: 0:12: 'assign': l-value required (can't modify a const variable)

Storage Qualifiers

217

Attribute Variables
As you have seen in previous chapters, attribute variables are available only in vertex
shaders. They must be declared as a global variable and are used to pass per-vertex data
to the vertex shader. You should note that it is “per-vertex.” For example, if there are two
vertices, (4.0, 3.0, 6.0) and (8.0, 3.0, 0.0), data for each vertex can be passed to an attribute
variable. However, data for other coordinates, such as (6.0, 3.0, 3.0), which is a halfway
point between the two vertices and not a specified vertex, cannot be passed to the variable. If you want to do that, you need to add the coordinates as a new vertex. Attribute
variables can only be used with the data types float, vec2, vec3, vec4, mat2, mat3, and
mat4. For example:
attribute vec4 a_Color;
attribute float a_PointSize;

There is an implementation-dependent limit on the number of attribute variables available, but the minimum number is 8. The limits on the number of each type of variable are
shown in Table 6.14.
Table 6.14

The Limitation on the Number of Attribute, Uniform, and Varying Variables
The Built-In Constants for the
Maximum Number

Minimum
Number

const mediump int
gl_MaxVertexAttribs

8

Vertex shader

const mediump int
gl_MaxVertexUniformVectors

128

Fragment shader

const mediump int
gl_MaxFragmentUniformVectors

16

const mediump int
gl_MaxVaryingVectors

8

Types of Variables
attribute variables
uniform variables

varying variables

Uniform Variables
Uniform variables are allowed to be used in both vertex and fragment shaders and must
be declared as global variables. Uniform variables are read-only and can be declared as any
data types other than array and structure. If a uniform variable of the same name and data
type is declared in both a vertex shader and a fragment shader, it is shared between them.
Uniform variables contain “uniform” (common) data, so your JavaScript program must
only use them to pass such data. For example, because transformation matrices contain
the uniform values for all vertices, they can be passed to uniform variables:
uniform mat4 u_ViewMatrix;
uniform vec3 u_LightPosition;

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There is an implementation-dependent limit on the number of uniform variables that can
be used (Table 6.14). Note that the limit in a vertex shader is different from that in a fragment shader.

Varying Variables
The last type of qualifier is varying. Varying variables also must be declared as global
variables and are used to pass data from a vertex shader to a fragment shader by declaring
a variable with the same type and name in both shaders. (See v_Color in Listing 6.1 and
Listing 6.2.) The following are examples of varying variable declarations:
varying vec2 v_TexCoord;
varying vec4 v_Color;

Just like attribute variables, the varying variables can be declared only with the following
data types: float, vec2, vec3, vec4, mat2, mat3, and mat4. As explained in Chapter 5, the
value of a varying variable written by a vertex shader is not passed to a fragment shader as
is. Rather, the rasterization process between the vertex and fragment shaders interpolates
the value according to the shape to be drawn, and then the interpolated value is passed
per fragment. This interpolation process is the reason for the limitations on the data types
that can be used with a varying variable.
The number of varying variables also has an implementation dependent limit. The
minimum number is 8 (see Table 6.14).

Precision Qualifiers
Precision qualifiers were newly introduced in GLSL ES to make it possible to execute
shader programs more efficiently and to reduce their memory size. As the name suggests,
it is a simple mechanism to specify how much precision (the number of bits) each data
type should have. Simply put, specifying higher precision data requires more memory
and computation time, and specifying lower precision requires less. By using these qualifiers, you can exercise fine-grained control over aspects of performance and size. However,
precision qualifiers are optional, and a reasonable default compromise can be specified
using the following lines:
#ifdef GL_ES
precision mediump float;
#endif

Because WebGL is based on OpenGL ES 2.0, which was designed for consumer electronics
and embedded systems, WebGL programs may end up executing on a range of hardware
platforms. In some cases, the computation time and memory efficiency could be improved
by using lower precision data types when performing calculations and operations. Perhaps
more importantly, this also enables reduced power consumption and thus extended
battery life on mobile devices.

Precision Qualifiers

219

You should note, however, that just specifying lower precision may lead to incorrect
results within WebGL, so it’s important to balance efficiency and correctness.
As shown in Table 6.15, WebGL supports three types of precision qualifiers: highp (high
precision), mediump (medium precision), and lowp (lower precision).
Table 6.15
Precision
Qualifiers

highp

mediump

lowp

Precision Qualifiers
Descriptions

High precision. The minimum precision
required for a vertex shader.
Medium precision. The minimum precision
required for a fragment shader. More than
lowp, and less than highp.
Low precision. Less than mediump, but all
colors can be represented.

Default Range and Precision
Float

int

(–262, 262)

(–216, 216)

Precision: 2–16
(–214, 214)

(–210, 210)

Precision: 2–10
(–2, 2)

(–28, 28)

Precision: 2–8

There are a couple of things to note. First, fragment shaders may not support highp in
some WebGL implementations; a way to check this is shown later in this section. Second,
the actual range and precision are implementation dependent, which you can check by
using gl.getShaderPrecisionFormat().
The following are examples of the declaration of variables using the precision qualifiers:
mediump float size; // float of medium precision
highp vec4 position; // vec4 composed of floats of high precision
lowp vec4 color;
// vec4 composed of floats of lower precision

Because specifying a precision for all variables is time consuming, a default for each data
type can be set using the keyword precision, which must be specified at the top of a
vertex shader or fragment shader using the following syntax:
precision precision-qualifier name-of-type;

This sets the precision of the data type specified by name-of-type to the precision specified
by precision-qualifier. In this case, variables declared without a precision qualifier have this
default precision automatically set. For example:
precision mediump float; // All floats have medium precision
precision highp int;
// All ints have high precision

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This specifies all data types related to float, such as vec2 and mat3, to have medium precision, and all integers to have high precision. For example, because vec4 consists of four
float types, each float of the vector is set to medium precision.
You may have noticed that in the examples in previous chapters, you didn’t specify precision qualifiers to the data types other than float in fragment shaders. This is because
most data types have a default precision value; however, there is no default precision for
float in a fragment shader. See Table 6.16 for details.
Table 6.16

Default Precision of Type

Type of Shader

Data Type

Default Precision

Vertex shader

int

highp

float

highp

sampler2D

lowp

samplerCube

lowp

int

medium

float

None

sampler2D

lowp

samplerCube

lowp

Fragment shader

The fact that there is no default precision for float requires programmers to carefully use
floats in their fragment shaders. So, for example, using a float without specifying the
precision will result in the following error:
failed to compile shader: ERROR: 0:1 : No precision specified for (float).

As mentioned, whether a WebGL implementation supports highp in a fragment shader is
implementation dependent. If it is supported, the built-in macro GL_FRAGMENT_PRECISION_
HIGH is defined (see the next section).

Preprocessor Directives
GLSL ES supports preprocessor directives, which are commands (directives) for the preprocessor stage before actual compilation. They are always preceded by a hash mark (#). The
following example was used in ColoredPoints.js:
#ifdef GL_ES
precision mediump float;
#endif

Preprocessor Directives

221

These lines check to see if the macro GL_ES is defined, and if so the lines between #ifdef
and #endif are executed. They are similar to an if statement in JavaScript or C.
The following three preprocessor directives are available in GLSL ES:
#if constant-expression
If the constant-expression is true, this part is executed.
#endif
#ifdef macro
If the macro is defined, this part is executed.
#endif
#ifndef macro
If the macro is not defined, this part is executed.
#endif

The #define is used to define macros. Unlike C, macros in GLSL ES cannot have macro
parameters:
#define macro-name string

You can use #undef to undefine the macro:
#undef macro-name

You can use #else directives just like an if statement in JavaScript or C. For example:
#define NUM 100
#if NUM == 100
If NUM == 100 then this part is executed.
#else
If NUM != 100 then this part is executed.
#endif

Macros can use any name except for the predefined macros names shown in Table 6.17.
Table 6.17

Predefined Macros

Macro

Description

GL_ES

Defined and set to 1 in OpenGL ES 2.0

GL_FRAGMENT_PRECISION_HIGH

highp is supported in a fragment shader

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So you can use the macro with preprocessor directives as follows:
#ifdef GL_ES
#ifdef GL_FRAGMENT_PRECISION_HIGH
precision highp float; // highp is supported. floats have high precision
#else
precision mediump float; // highp is not supported. floats have medium precision
#endif
#endif

You can specify which version of GLSL ES is used in the shader by using the #version
directive:
#version number

Accepted versions include 100 (for GLSL ES 1.00) and 101 (for GLSL ES 1.01). By default,
shaders that do not include a #version directive will be treated as written in GLSL ES
version 1.00. The following example specifies version 1.01:
#version 101

The #version directive must be specified at the top of the shader program and can only be
preceded by comments and white space.

Summary
This chapter explained the core features of the OpenGL ES Shading Language (GLSL ES) in
some detail.
You have seen that the GLSL ES shading language has many similarities to C but has
been specialized for computer graphics and has had unnecessary C features removed. The
specialized computer graphics features include support for vector and matrix data types,
special component names for accessing the components of a vector or matrix, and operators for a vector or matrix. In addition, GLSL ES supports many built-in functions for operations frequently used in computer graphics, all designed to allow you to create efficient
shader programs.
Now that you have a better understanding of GLSL ES, the next chapter will return to
WebGL and explore more sophisticated examples using this new knowledge.

Summary

223

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Chapter 7

Toward the 3D World

In the previous chapters, we used examples based on 2D geometrical shapes
to explain how the WebGL system works and how it supports the behavior
of shaders, transformations such as translation and rotation, animation, and
texture mapping. However, the techniques you’ve learned so far can be applied
not only to 2D shapes but also to 3D objects. In this chapter you’ll take the first
step into the 3D world and explore the implications of moving from 2D to 3D.
In particular, you will explore:

• Representing the user’s view into the 3D world
• Controlling the volume of 3D space that is viewed
• Clipping
• Handling foreground and background objects
• Drawing a 3D object (a cube)
All these issues have a significant impact on how the 3D scene is drawn and
presented to viewers, and a mastery of them is critical to building compelling 3D scenes. As usual, we’ll take you step by step so you will quickly master
the basics and be able to move on to the more complex issues of lighting and
performance in the final chapters.

What’s Good for Triangles Is Good for Cubes
So far, you’ve used the humble triangle in many of the explorations and
programs. As previously discussed, you’ve seen how 3D objects are composed
of 2D shapes—in particular the triangle. Figure 7.1 shows a cube that has been
built up from 12 triangles.

Figure 7.1

A cube composed of triangles

So when you deal with 3D objects, you just need to apply the techniques you have
learned to each triangle that makes up the objects. The only difference from past examples, and it’s a significant one, is that you now need to consider the depth information
of the triangles in addition to the x and y coordinates. Let’s begin by exploring how you
specify and control the viewing direction—that is, the view into the 3D scene the user
has—and then look at the visible range that controls how much of the scene the user sees.
The explanations focus on the basic triangle because it simplifies things; however, what’s
true for triangles is true for 3D objects.

Specifying the Viewing Direction
The critical factor when considering 3D objects is that they have depth in a 3D space.
This means you need to take care of several issues that you didn’t have to consider when
using 2D shapes. First, because of the nature of 3D space, you can look at the object from
anywhere in the space; that is, your viewpoint can be anywhere. When describing the way
you view objects, two important points need consideration:

• The viewing direction (where you are looking from, and at which part of the scene
are you looking?)

• The visible range (given the viewing direction, where can you actually see?)
In this first section let’s explore viewing direction and the techniques that allow you to
place the eye point anywhere in 3D space and then look at objects from various directions. You’ll take a look at the second item and see how to specify the visible range in the
next section.
As introduced in Chapter 2, “Your First Step with WebGL” (refer to Figure 2.16), let’s
assume that, by default, the eye point is placed at the origin (0, 0, 0), and the line of sight
extends along the z-axis in the negative direction (inward toward your computer screen).
In this section, you will move the eye point from the default location to other locations
and then view a triangle from there.

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Let’s construct a sample program, LookAtTriangles, that locates the eye point at (0.20,
0.25, 0.25) and then views three triangles from there toward the origin (0, 0, 0). Using
three triangles makes it easy to understand the depth information in the 3D scene. Figure
7.2 shows a screen shot of LookAtTriangles and the color and z coordinate of each
triangle.

y

-0.75
0.75

x

z
Figure 7.2

LookAtTriangles (left), and the color and z coordinate of each triangle (right)

The program uses softer colors because they are easier on the eyes.

Eye Point, Look-At Point, and Up Direction
To specify where you are looking from and which part of the scene you are looking at in
the 3D space, you need two items of position information: the eye point (where you are
looking from) and the look-at point (which part of the scene you are looking at). In addition, in 3DCG, you need to specify which direction is up in the scene. As such, a total of
three items of information are required to specify the viewing direction (see Figure 7.3).

up vector

(u p X , u p Y , u p Z )

e y e p o in t
lo o k -a t p o in t
( a tX , a tY , a tZ )

scene

(e y e X , e y e Y , e y e Z )

Figure 7.3

Eye point, look-at point, and up direction

Specifying the Viewing Direction

227

Eye point: This is the starting point from which the 3D space is viewed. In the following
sections, the coordinates of this position are referred to as (eyeX, eyeY, eyeZ).
Look-at point: This is the point at which you are looking and which determines the
direction of the line of sight from the eye point. As the name suggests, the eye point is a
point, not a vector, so another point (such as the look-at point) is required to determine
the direction in which you are looking. The look-at point is a point on the line of sight
extending from the eye point. The coordinates of the look-at point are referred to as
(atX, atY, atZ).
Up direction: This determines the up direction in the scene that is being viewed from
the eye point to the look-at point. If only the eye point and the look-at point are determined, there is freedom to rotate the line of sight from the eye point to the look-at
point. (In Figure 7.4, inclining the head causes the top and bottom of the scene to shift.)
To define the rotation, you must determine the up direction along the line of sight. The
up direction is specified by three numbers representing the direction. The coordinates for
this direction are referred to as (upX, upY, upZ).
up vector
eye point
lo o k -a t p o in t
scene

Figure 7.4

Eye point, look-at point, and up direction

In WebGL, you can specify the position and direction the eye point faces by converting
these three items of information in a matrix and passing the matrix to a vertex shader.
This matrix is called a view transformation matrix or view matrix, because it changes
the view of the scene. In cuon-matrix.js, the method Matrix4.setLookAt() is defined
to calculate the view matrix from the three items of information: eye point, look-at point
and up direction.

Matrix4.setLookAt(eyeX, eyeY, eyeZ, atX, atY, atZ, upX, upY, upZ)
Calculate the view matrix derived from the eye point (eyeX, eyeY, eyeZ), the look-at point
(atX, atY, atZ), and the up direction (upX, upY, upZ). This view matrix is set up in the
Matrix4 object. The look-at point is mapped to the center of the .
Parameters

228

eyeX, eyeY, eyeZ

Specify the position of the eye point.

atX, atY, atZ

Specify the position of the look-at point.

CHAPTER 7

Toward the 3D World

upX, upY, upZ
Return value

Specify the up direction in the scene. If the up direction is
along the positive y-axis, then (upX, upY, upZ) is (0, 1, 0).

None

In WebGL, the default settings when using Matrix4.setLookAt() are defined as follows:

• The eye point is placed at (0, 0, 0) (that is, the origin of the coordinate system).
• The look-at point is along the negative z-axis, so a good value is (0, 0, –1).1 The up
direction is specified along the positive y-axis, so a good value is (0, 1, 0).
So, for example, if the up direction is specified as (1, 0, 0), the positive x-axis becomes the
up direction; in this case, you will see the scene tilted by 90 degrees.
A view matrix representing the default settings in WebGL can be simply produced as
follows (see Figure 7.5).
v a r i n itia lV ie w M a tr ix = n e w M a tr ix 4 ( ) ;
in itia lV ie w M a tr ix .s e tL o o k A t( 0 , 0 , 0 , 0 , 0 , - 1 , 0 , 1 , 0 ) ;

e y e p o in t

Figure 7.5

lo o k - a t p o in t

u p v e c to r

An example of setLookAt()

Now you understand how to use the method setLookAt(), so let’s take a look at its use in
an actual sample program.

Sample Program (LookAtTriangles.js)
LookAtTriangles.js, shown in Listing 7.1, is a program that changes the position of the

eye point and then draws the three triangles shown in Figure 7.2. Although it is difficult
to see on paper, the three triangles are, in order of proximity, blue, yellow, and green,
respectively, all fading to red in the bottom-right corner.
Listing 7.1

LookAtTriangles.js

1 // LookAtTriangles.js
2 // Vertex shader program
3 var VSHADER_SOURCE =
4
'attribute vec4 a_Position;\n' +
5
'attribute vec4 a_Color;\n' +
6
'uniform mat4 u_ViewMatrix;\n' +

1

The z component could be any negative value. The value -1 is an example but we could have chosen
any other negative value.

Specifying the Viewing Direction

229

7
'varying vec4 v_Color;\n' +
8
'void main() {\n' +
9
' gl_Position = u_ViewMatrix * a_Position;\n' +
10
' v_Color = a_Color;\n' +
11
'}\n';
12
13 // Fragment shader program
14 var FSHADER_SOURCE =
...
18
'varying vec4 v_Color;\n' +
19
'void main() {\n' +
20
' gl_FragColor = v_Color;\n' +
21
'}\n';
22
23 function main() {
...
40
// Set the vertex coordinates and color (blue triangle is in front)
41
var n = initVertexBuffers(gl);
...
50
// Get the storage location of u_ViewMatrix variable
51
var u_ViewMatrix = gl.getUniformLocation(gl.program,'u_ViewMatrix');
...
57
// Set the eye point, look-at point, and up direction
58
var viewMatrix = new Matrix4();
59
viewMatrix.setLookAt(0.20, 0.25, 0.25, 0, 0, 0, 0, 1, 0);
60
61
// Pass the view matrix to u_ViewMatrix variable
62
gl.uniformMatrix4fv(u_ViewMatrix, false, viewMatrix.elements);
...
67
// Draw a triangle
68
gl.drawArrays(gl.TRIANGLES, 0, n);
69 }
70
71 function initVertexBuffers(gl) {
72
var verticesColors = new Float32Array([
73
// vertex coordinates and color
74
0.0, 0.5, -0.4, 0.4 1.0, 0.4, // The back green triangle
75
-0.5, -0.5, -0.4, 0.4 1.0, 0.4,
76
0.5, -0.5, -0.4, 1.0, 0.4 0.4,
77
78
0.5, 0.4, -0.2, 1.0, 0.4 0.4, // The middle yellow triangle
79
-0.5, 0.4, -0.2, 1.0, 1.0, 0.4,
80
0.0, -0.6, -0.2, 1.0, 1.0, 0.4,
81
82

230

0.0,

0.5,

CHAPTER 7

0.0,

0.4

0.4

1.0,

Toward the 3D World

// The front blue triangle

83
84
85
86
87
88
89

-0.5, -0.5,
0.0, 0.4 0.4 1.0,
0.5, -0.5,
0.0, 1.0, 0.4 0.4
]);
var n = 9;

96
97

// Create a buffer object
var vertexColorbuffer = gl.createBuffer();
...
gl.bindBuffer(gl.ARRAY_BUFFER, vertexColorbuffer);
gl.bufferData(gl.ARRAY_BUFFER, verticesColors, gl.STATIC_DRAW);
...
return n;

121
122 }

This program is based on ColoredTriangle.js in Chapter 5, “Using Colors and Texture
Images.” The fragment shader, the method of passing the vertex information, and so on,
is the same as in ColoredTriangle.js. The three main differences follow:

• The view matrix is passed to the vertex shader (line 6) and then multiplied by the
vertex coordinates (line 9).

• The vertex coordinates and color values of the three triangles (line 72 to 85) are set
up in initVertexBuffers(), which is called from line 41 of main() in JavaScript.

• The view matrix is calculated at lines 58 and 59 in main() and passed to the uniform
variable u_ViewMatrix in the vertex shader at line 62. You should note that the position of the eye point is (0.2, 0.25, 0.25); the position of the look-at point is (0, 0, 0);
the up direction is (0, 1, 0).
Let’s start by looking at the second difference and the function initVertexBuffers() (line
71). The difference between this program and the original program, ColoredTriangle.
js, is that verticesColors at line 72 (which is the array of vertex coordinates and colors
for a single triangle) is modified for the three triangles, and the z coordinates are added
in the array. These coordinates and colors are stored together in the buffer object vertexColorBuffer (lines 96 and 97) created at line 89. Because you are now dealing with three
triangles (each with three vertices), you need to specify 9 as the third argument of gl.
drawArrays() at line 68.
To specify the view matrix (that is, where you are looking and which part of the scene you
are looking at [item 3]), you need to set up and pass the view matrix to the vertex shader.
To do this, a Matrix4 object viewMatrix is created at line 58, and you use setLookAt() to
calculate and store the view matrix to viewMatrix at line 59. This view matrix is passed to
u_ViewMatrix at line 62, which is the uniform variable used in the vertex shader:
57
58
59
60

// Set the eye point, look-at point, and up direction
var viewMatrix = new Matrix4();
viewMatrix.setLookAt(0.20, 0.25, 0.25, 0, 0, 0, 0, 1, 0);

Specifying the Viewing Direction

231

61
62

// Pass the view matrix to the u_ViewMatrix variable
gl.uniformMatrix4fv(u_ViewMatrix, false, viewMatrix.elements);

Those are all the changes needed in the JavaScript program. Now let’s examine what is
happening in the vertex shader:
2 // Vertex shader program
3 var VSHADER_SOURCE =
4
'attribute vec4 a_Position;\n' +
5
'attribute vec4 a_Color;\n' +
6
'uniform mat4 u_ViewMatrix;\n' +
7
'varying vec4 v_Color;\n' +
8
'void main() {\n' +
9
' gl_Position = u_ViewMatrix * a_Position;\n' +
10
' v_Color = a_Color;\n' +
11
'}\n';

The vertex shader starts from line 4. The only two lines that differ from ColoredTriangle.
js are indicated by boldface: Line 6 defines the uniform variable u_ViewMatrix, and line 9
multiplies the matrix by the vertex coordinates. These modifications seem quite trivial, so
how do they change the position of the eye point?

Comparing LookAtTriangles.js with RotatedTriangle_Matrix4.js
Looking at the vertex shader in this sample program, you may notice a similarity
with that in RotatedTriangle_Matrix4.js, which was explained in Chapter 4, “More
Transformations and Basic Animation.” That vertex shader created a rotation matrix using
a Matrix4 object and then used the matrix to rotate a triangle. Let’s take a look at that
shader again:
1 // RotatedTriangle_Matrix4.js
2 // Vertex shader program
3 var VSHADER_SOURCE =
4
'attribute vec4 a_Position;\n' +
5
'uniform mat4 u_rotMatrix;\n' +
6
'void main() {\n' +
7
' gl_Position = u_rotMatrix * a_Position;\n' +
8
'}\n';

The vertex shader in this section (LookAtTriangles.js) is listed as follows:
1 // LookAtTriangles.js
2 // Vertex shader program
3 var VSHADER_SOURCE =
4
'attribute vec4 a_Position;\n' +
5
'attribute vec4 a_Color;\n' +
6
'uniform mat4 u_ViewMatrix;\n' +

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7
8
9
10
11

'varying vec4 v_Color;\n' +
'void main() {\n' +
' gl_Position = u_ViewMatrix * a_Position;\n' +
' v_Color = a_Color;\n' +
'}\n';

As you can see, the attribute variable for color values (a_Color) and the varying variable
that passes the values to the fragment shader (v_Color) were added, and the name of the
uniform variable was changed from u_rotMatrix to u_ViewMatrix. Despite these differences, the calculation of the value assigned to gl_Position is the same as that in the
vertex shader of RotatedTriangle_Matrix4.js: multiplying a mat4 matrix by a_Position.
(Compare line 7 in RotatedTriangle_Matrix4.js with line 9 in LookAtTriangles.js.)
This tells you that the operation performing “where you are looking and which part of the
scene you are looking at” is actually equivalent to transformations such as translating or
rotating a triangle.
Let’s use an example to explain this. Assume that you are looking at a triangle from the
origin (0, 0, 0) along the negative direction of the z-axis, and then the eye point moves to
the position (0, 0, 1) (the left-side figure of Figure 7.6). In this case, the distance between
the eye point and the triangle has increased by 1.0 unit of the z-axis. To achieve the same
effect, you could leave the eye point alone and instead move the triangle 1.0 unit away
(the right-side figure of Figure 7.6).

y

y

e y e p o in t

z

Figure 7.6

x

z

x

Movement of the eye point is identical to that of objects in the scene

This is exactly what happens in our sample program. The setLookAt() method of the
Matrix4 object just calculates the matrix to carry out this transformation using the information about the position of the eye point, look-at point, and up direction. So, by then
multiplying the matrix by the vertex coordinates of the objects in the scene, you obtain
the same effect as moving the eye point. Essentially, instead of moving the eye point
in one direction, the objects viewed (that is, the world itself) are moved in the opposite
direction. You can use the same approach to handle rotation of the eye point.
Because moving the eye point is the same type of transformation as rotating or translating
a triangle, you can represent both of them as a transformation matrix. Let’s look at how

Specifying the Viewing Direction

233

you calculate the matrix when you want to rotate a triangle and move the position of the
eye point.

Looking at Rotated Triangles from a Specified Position
RotatedTriangle_Matrix4 in Chapter 4 displayed the triangle rotated around the z-axis. In
this section, you modify LookAtTriangles to display the triangles viewed from a specified

eye point along the line of sight. In this case, two matrices are required: a rotation matrix
to rotate the triangles and a view matrix to specify the view of the scene. The first issue to
consider is in which order you should multiply them.
So far, you know that multiplying a matrix by a vertex coordinate will apply the transformation defined by the matrix to the coordinates. That is to say, multiplying a rotation
matrix by a vertex coordinate causes it to be rotated.
Multiplying a view matrix by a vertex coordinate causes the vertex to be transformed to
the correct position as viewed from the eye position. In this sample program, we want to
view the rotated triangles from a specified position, so we need to rotate the triangles and
then look at them from the specified eye position. In other words, we need to rotate the
three vertex coordinates comprising the triangle. Then we need to transform the rotated
vertex coordinates (the rotated triangle) as we look at them from the specified position.
We can achieve this by carrying out a matrix multiplication in the order described in the
previous sentence. Let’s check the equations.
As explained previously, if you want to rotate a shape, you need to multiply a rotation
matrix by the vertex coordinates of the shape as follows:
〈" rotated " vertex coordinates 〉 =
〈rotation matrix 〉 × 〈original vertex coordinates 〉
By multiplying a view matrix by the rotated vertex coordinates in the preceding equation,
you can obtain the rotated vertex coordinates that are viewed from the specified position.
〈" rotated " vertex coordinates " viewed from specified position "〉 =
〈view matrix 〉 × 〈" rotated " vertex coordinates 〉
If you substitute the first expression into the second one, you obtain the following:
〈" rotated " vertex coordinates " viewed from specified position " 〉 =
〈view matrix 〉 × 〈rotation matrix 〉 × 〈original vertex coordinates 〉
In this expression, you use a rotation matrix, but you can also apply a translation matrix,
a scaling matrix, or a combination of them. Such a matrix is generally called a model
matrix. Using that term, you can rewrite the expression shown in Equation 7.1.
Equation 7.1
〈view matrix 〉 × 〈 model matrix 〉 × 〈vertex coordinates 〉

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Now you need to implement this expression in a shader program, but because it is quite
a simple expression, you can implement it as-is in a vertex shader. The sample program
LookAtRotatedTriangles implements the transformation, and a screen shot is shown in
Figure 7.7. Note, in this figure, that the white dashed line shows the triangle before rotation so that you can easily see the rotation.

Figure 7.7

LookAtRotatedTriangles

Sample Program (LookAtRotatedTriangles.js)
LookAtRotatedTriangles.js is programmed by slightly modifying LookAtTriangles.js.
You just need to add the uniform variable u_ModelMatrix to pass the model matrix to the
shader and then add some processing in JavaScript’s main() function to pass the matrix to
the variable. The relevant code is shown in Listing 7.2.

Listing 7.2

LookAtRotatedTriangles.js

1 // LookAtRotatedTriangles.js
2 // Vertex shader program
3 var VSHADER_SOURCE =
4
'attribute vec4 a_Position;\n' +
5
'attribute vec4 a_Color;\n' +
6
'uniform mat4 u_ViewMatrix;\n' +
7
'uniform mat4 u_ModelMatrix;\n' +
8
'varying vec4 v_Color;\n' +
9
'void main() {\n' +
10 ' gl_Position = u_ViewMatrix * u_ModelMatrix * a_Position;\n' +
11 ' v_Color = a_Color;\n' +
12 '}\n';
...

Specifying the Viewing Direction

235

24 function main() {
...
51
// Get the storage locations of u_ViewMatrix and u_ModelMatrix
52
varu_ViewMatrix = gl.getUniformLocation(gl.program,'u_ViewMatrix');
53
var u_ModelMatrix = gl.getUniformLocation(gl.program, 'u_ModelMatrix');
...
59
// Specify the eye point and line of sight
60
var viewMatrix = new Matrix4();
61
viewMatrix.setLookAt(0.20, 0.25, 0.25, 0, 0, 0, 0, 1, 0);
62
63
// Calculate the rotation matrix
64
var modelMatrix = new Matrix4();
65
modelMatrix.setRotate(-10, 0, 0, 1); // Rotate around z-axis
66
67
// Pass each matrix to each uniform variable
68
gl.uniformMatrix4fv(u_ViewMatrix, false, viewMatrix.elements);
69
gl.uniformMatrix4fv(u_ModelMatrix, false, modelMatrix.elements);

First, let’s examine the vertex shader. You can see that line 10 simply implements
Equation 7.1 as-is by using u_ModelMatrix at line 7, which receives data from the
JavaScript program:
10

'

gl_Position = u_ViewMatrix * u_ModelMatrix * a_Position;\n' +

In the main() function in JavaScript, you already have the code for calculating a view
matrix, so you just need to add the code for calculating the rotation matrix performing a –10 degree rotation around the z-axis. Line 53 gets the storage location of the
u_ModelMatrix variable, and line 64 creates modelMatrix for the rotation matrix. Then
line 65 calculates the matrix using Matrix4.setRotate(), and line 69 passes it to
u_ModelMatrix.
When you run this sample program, you will see the triangles shown in Figure 7.6 illustrating that the matrices multiplied by the vertex coordinates (a_Position) have the
desired effect. That is, the vertex coordinates were rotated by u_ModelMatrix, and then
the resulting coordinates were transformed by u_ViewMatrix to the correct position as if
viewed from the specified position.

Experimenting with the Sample Program
In LookAtRotatedTriangles.js, you implemented Equation 7.1 as- is. However, because
the multiplication of the view matrix and model matrix is performed per vertex in the
vertex shader, this implementation is inefficient when processing many vertices. The
result of the matrix multiplication in Equation 7.1 is identical for each vertex, so you can
calculate it in advance and pass the result to the vertex shader. The matrix obtained by
multiplying a view matrix by a model matrix is called a model view matrix. That is,

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〈 model view matrix 〉 = 〈view matrix 〉 × 〈 model matrix 〉
Then, you can rewrite the expression in Equation 7.1 as shown in Equation 7.2.
Equation 7.2
〈 model view matrix 〉 × 〈vertex coordinates 〉
If you use Equation 7.2, you can rewrite the sample program shown in Listing 7.3. This
sample program is LookAtRotatedTriangles_mvMatrix.
Listing 7.3

LookAtRotatedTriangles_mvMatrix.js

1 // LookAtRotatedTriangles_mvMatrix.js
2 // Vertex shader program
3 var VSHADER_SOURCE =
...
6
'uniform mat4 u_ModelViewMatrix;\n' +
7
'varying vec4 v_Color;\n' +
8
'void main() {\n' +
9
' gl_Position = u_ModelViewMatrix * a_Position;\n' +
10
' v_Color = a_Color;\n' +
11
'}\n';
...
23 function main() {
...
50
// Get the storage locations of u_ModelViewMatrix and u_ModelMatrix
51
var u_ModelViewMatrix = gl.getUniformLocation(gl.program, 'u_ModelViewMatrix');
...
59
viewMatrix.setLookAt(0.20, 0.25, 0.25, 0, 0, 0, 0, 1, 0);
...
63
modelMatrix.setRotate(-10, 0, 0, 1); // Calculate rotation matrix
64
65
// Multiply both matrices
66
var modelViewMatrix = viewMatrix.multiply(modelMatrix);
67
68
// Pass the model viewmatrix to u_ModelViewMatrix
69
gl.uniformMatrix4fv(u_ModelViewMatrix, false, modelViewMatrix.elements);

In the vertex shader, the name of the uniform variable was modified to u_
ModelViewMatrix and calculated in line 9. However, the processing steps in the vertex
shader are identical to the original LookAtTriangles.js.
Within the JavaScript program, the method of calculating viewMatrix and modelMatrix
from lines 59 to 63 is identical to that in LookAtRotatedTriangles.js and, when multiplied, result in modelViewMatrix (line 66). The multiply() method is used to multiply
Specifying the Viewing Direction

237

Matrix4 objects. It multiplies the matrix on the right side (viewMatrix) by the matrix
specified by the argument (modelMatrix) of the method. So this code actually performs
modelViewMatrix= viewMatrix * modelMatrix. Unlike with GLSL ES, you need to use a

method to perform matrix multiplication instead of the * operator.
Having obtained modelViewMatrix, you just need to pass it to the u_ModelViewMatrix
variable at line 69. Once you run the program, you can see the same result as shown in
Figure 7.6.
As a final point, in this sample program, each matrix was calculated piece by piece at lines
59, 63, and 66 to better show the flow of the calculation. However, this could be rewritten
in one line for efficiency:
var modelViewMatrix = new Matrix4();
modelViewMatrix.setLookAt(0.20, 0.25, 0.25, 0, 0, 0, 0, 1, 0).rotate(-10, 0, 0, 1);
// Pass the model view matrix to the uniform variable
gl.uniformMatrix4fv(u_ModelViewMatrix, false, modelViewMatrix.elements);

Changing the Eye Point Using the Keyboard
Let’s modify LookAtTriangles to change the position of the eye point when the arrow
keys are pressed. LookAtTrianglesWithKeys uses the right arrow key to increase the x coordinate of the eye point by 0.01 and the left arrow key to decrease the coordinate by 0.01.
Figure 7.8 shows a screen shot of the sample program when run. If you hold down the left
arrow key, the scene changes to that seen on the right side of Figure 7.8.

Figure 7.8

LookAtTrianglesWithKeys

Sample Program (LookAtTrianglesWithKeys.js)
Listing 7.4 shows the sample code. The vertex shader and the fragment shader are the
same as those in LookAtTriangles.js. The basic processing flow of main() in JavaScript
is also the same. The code for registering the event handler called on a key press is added

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to the sample program, and the code for drawing the triangles has been moved into the
function draw().
Listing 7.4

LookAtTrianglesWithKeys.js

1 // LookAtTrianglesWithKeys.js
2 // Vertex shader program
3 var VSHADER_SOURCE =
4
'attribute vec4 a_Position;\n' +
5
'attribute vec4 a_Color;\n' +
6
'uniform mat4 u_ViewMatrix;\n' +
7
'varying vec4 v_Color;\n' +
8
'void main() {\n' +
9
' gl_Position = u_ViewMatrix * a_Position;\n' +
10
' v_Color = a_Color;\n' +
11
'}\n';
...
23 function main() {
...
50
// Get the storage location of the u_ViewMatrix variable
51
varu_ViewMatrix = gl.getUniformLocation(gl.program,'u_ViewMatrix');
...
57
// Create Matrix4 object for a view matrix
58
var viewMatrix = new Matrix4();
59
// Register the event handler to be called on key press
60
document.onkeydown = function(ev){ keydown(ev, gl, n, u_ViewMatrix,
➥viewMatrix); };
61
62
draw(gl, n, u_ViewMatrix, viewMatrix); // Draw a triangle
63 }
...
117 var g_eyeX = 0.20, g_eyeY = 0.25, g_eyeZ = 0.25; // The eye point
118 function keydown(ev, gl, n, u_ViewMatrix, viewMatrix) {
119
if(ev.keyCode == 39) { // The right arrow key was pressed
120
g_eyeX += 0.01;
121
} else
122
if (ev.keyCode == 37) { // The left arrow key was pressed
123
g_eyeX -= 0.01;
124
} else { return; } // Prevent unnecessary drawing
125
draw(gl, n, u_ViewMatrix, viewMatrix);
126 }
127
128 function draw(gl, n, u_ViewMatrix, viewMatrix) {
129
// Set the eye point and line of sight
130
viewMatrix.setLookAt(g_eyeX, g_eyeY, g_eyeZ, 0, 0, 0, 0, 1, 0);

Specifying the Viewing Direction

239

131
132
133
134
135
136
137
138 }

// Pass the view matrix to the u_ViewMatrix variable
gl.uniformMatrix4fv(u_ViewMatrix, false, viewMatrix.elements);
gl.clear(gl.COLOR_BUFFER_BIT); // Clear 
gl.drawArrays(gl.TRIANGLES, 0, n); // Draw a triangle

In this sample, you are using the event handler to change the position of the eye point
when the right arrow key or the left arrow key is pressed. Before explaining the event
handler, let’s look at the function draw() that is called from the event handler.
The process performed in draw() is straightforward. Line 130 calculates the view matrix
using the global variables g_eyeX, g_eyeY, and g_eyeZ defined at line 117, which contain
0.2, 0.25, and 0.25, respectively. Then the matrix is passed to the uniform variable
u_ViewMatrix in the vertex shader at line 133. Back in main(), the storage location of
u_ViewMatrix is retrieved at line 51, and a Matrix4 object (viewMatrix) is created at line
58. These two operations are carried out in advance because it is redundant if you perform
them for each draw operation, particularly retrieving the storage. After that, line 135 clears
, and line 137 draws the triangles.
The variables g_eyeX, g_eyeY, and g_eyeZ specify the eye position and are recalculated
in the event handler whenever a key is pressed. To call the event handler on key press,
you need to register it to the onkeydown property of the document object. In this event
handler, because you need to call draw() to draw the triangles, you must pass all arguments that draw() requires. This is why an anonymous function registers the handler as
follows:
59
60

// Register the event handler to be called on key press
document.onkeydown = function(ev){ keydown(ev, gl, n, u_ViewMatrix,
➥viewMatrix); };

This sets up the event handler keydown() to be called when the key is pressed. Let’s
examine how keydown() is implemented.
118 function keydown(ev, gl, n, u_ViewMatrix, viewMatrix) {
119
if(ev.keyCode == 39) { // The right arrow key was pressed
120
g_eyeX += 0.01;
121
} else
122
if (ev.keyCode == 37) { // The left arrow key was pressed
123
g_eyeX -= 0.01;
124
} else { return ; }
// Prevent unnecessary drawing
125
draw(gl, n, u_ViewMatrix, viewMatrix);
// Draw a triangle
126 }

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What keydown() is doing is also straightforward. When a key is pressed, keydown()
is called with the information about the event stored in the first parameter, ev, of
keydown(). Then you just need to check which key was pressed by examining the value of
ev.keyCode, modify g_eyeX, and draw the triangles. When the right arrow key is pressed,
the code increases g_eyeX by 0.01, and when the left arrow key is pressed, it decreases
g_eyeX by 0.01.
If you run the sample program, you can see the triangles shift every time you press the
arrow key.

Missing Parts
As you play with the sample program, you may notice that as you shift the eye position to
the extreme right or left, part of the triangle disappears (see Figure 7.9).

Figure 7.9

Part of the triangle disappears

This is because you haven’t specified the visible range (the boundaries of what you can
actually see) correctly. As mentioned in the first section of this chapter, WebGL does not
display objects outside the visible range. In the case of Figure 7.9, part of the triangle went
out of the visible range while pressing the arrow keys.

Specifying the Visible Range (Box Type)
Although WebGL allows you to place 3D objects anywhere in 3D space, it only displays
those that are in the visible range. In WebGL, this is primarily a performance issue; there’s
no point in drawing 3D objects if they are not visible to the viewer. In a way, this mimics
the way human sight works (see Figure 7.10); we see objects within the visible range based

Specifying the Visible Range (Box Type)

241

on our line of sight, which is approximately 200 degrees in the horizontal field of view.
WebGL also has a similar limited range and does not display 3D objects outside of that
range.

Figure 7.10

Human visual field

In addition to the up/down, left/right range along the line of sight, WebGL has a depth
range that indicates how far you can see. These ranges are called the viewing volume. In
Figure 7.9, because the depth range was not sufficient, part of the triangle disappears as it
moves out of the viewing volume.

Specify the Viewing Volume
There are two ways of specifying a viewing volume:

• Using a rectangular parallelepiped, or more informally, a box (orthographic projection)

• Using a quadrangular pyramid (perspective projection)
Perspective projection gives more information about depth and is often easier to view
because you use perspective views in real life. You should use this projection to show
the 3D scene in perspective, such as a character or a battlefield in a 3D shooting game.
Orthographic projection makes it much easier to compare two objects, such as two parts
of a molecule model, because there is no question about how the viewpoint may affect
the perception of distance. You should use the projection to show 3D objects in an orthographic view like those in technical drawing.
First, we will explain how the viewing volume works based on the box-shaped viewing
volume.
The box-shaped viewing volume is shaped as shown in Figure 7.11. This viewing volume
is set from the eye point toward the line of sight and occupies the space delimited by the
two planes: the near clipping plane and the far clipping plane. The near clipping plane
is defined by (right, top, -near), (-left, top, -near), (-left, -bottom, -near), and (right, -bottom,
-near). The far clipping plane is defined by (right, top, -far), (-left, top, -far), (-left, -bottom,
-far), and (right, -bottom, -far).

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far clipping plane

invisible

visible

near clipping plane

y
top

left

eye point

invisible

right
bottom

x

far

near

z
Figure 7.11

Box-shaped viewing volume

The scene viewed from the near clipping plane toward the line of sight is displayed on the
. If the aspect ratio of the near clipping plane is different from that of ,

the scene is scaled according to the ratio, and the aspect ratio of the geometric shapes or
objects in the scene are distorted. (You will explore this behavior in the last part of this
section.) The range from the near clipping plane to the far clipping plane defines the
viewing volume. Only objects inside this volume are displayed. If the objects lie partially
inside the volume, only the part inside the volume is displayed.

Defining a Box-Shaped Viewing Volume
To set the box-shaped viewing volume, you use the method setOrtho() supported by the
Matrix4 object defined in cuon-matrix.js.

Matrix4.setOrtho(left, right, bottom, top, near, far)
Calculate the matrix (orthographic projection matrix) that defines the viewing volume
specified by its arguments, and store it in Matrix4. However, left must not be equal to
right, bottom not equal to top, and near not equal to far.
Parameters

Return value

left, right

Specify the distances to the left side and right side of the near
clipping plane.

bottom, top

Specify the distances to the bottom and top of the near clipping
plane.

near, far

Specify the distances to the near and far clipping planes along
the line of sight.

None

Specifying the Visible Range (Box Type)

243

Here, you are using a matrix again, which in this case is referred to as the orthographic projection matrix. The sample program OrthoView will use this type of
projection matrix to set the box-shaped viewing volume and then draw three triangles—as used in LookAtRotatedTriangles—to test the effect of the viewing volume. In
LookAtRotatedTriangles, you placed the eye point at a different location from that of the
origin. However, in this sample program, you’ll use the origin (0, 0, 0) and set the line of
sight along the negative z-axis to make it easy to check the effect of the viewing volume.
The viewing volume is specified as shown in Figure 7.12, which uses near=0.0, far=0.5,
left=–1.0, right=1.0, bottom=–1.0, and top=1.0 because the triangles lie between 0.0 and –0.4
along the z-axis (refer to Figure 7.2).
y

(-1.0, 1.0, -0.5)

(-1.0, 1.0, 0.0)

(1.0, 1. 0, -0.5 )

e y e p o in t
(1.0, 1. 0, 0.0)

z
(-1.0, -1.0, -0. 5 )

(-1.0, -1.0, 0.0)

x
(1.0, -1.0, 0.0)

Figure 7.12

The box-shaped viewing volume used in OrthoView

In addition, we add key-press event handlers to change the values of near and far to check
the effect of changing the size of the viewing volume. The following are the active keys
and their mappings.
Arrow Key

Action

Right

Increases near by 0.01

Left

Decreases near by 0.01

Up

Increases far by 0.01

Down

Decreases far by 0.01

So that you can see the current values of near and far, they are displayed below the
canvas, as shown in Figure 7.13.

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Figure 7.13

OrthoView

Let’s examine the sample program.

Sample Program (OrthoView.html)
Because this sample program shows the near and far value on the web page and not in
the , you need to add something to the HTML file listing as shown in Listing 7.5
(OrthoView.html).
Listing 7.5

OrthoView.html

1 
2 
3

4

5

6

7
8

9

10
Please use a browser that supports 
11

12
The near and far values are displayed here.
13
14

...
18


Specifying the Visible Range (Box Type)

245

19

20 

As you can see, line 12 was added. This line shows “The near and far values are displayed
here” and uses JavaScript to rewrite the contents of nearFar to show the current near and
far values.

Sample Program (OrthoView.js)
Listing 7.6 shows OrthoView.js. This program is almost the same as
LookAtTrianglesWithKeys.js, which changes the position of the eye point by using the
arrow keys.
Listing 7.6

OrthoView.js

1 // OrthoView.js
2 // Vertex shader program
3 var VSHADER_SOURCE =
4
'attribute vec4 a_Position;\n' +
5
'attribute vec4 a_Color;\n' +
6
'uniform mat4 u_ProjMatrix;\n' +
7
'varying vec4 v_Color;\n' +
8
'void main() {\n' +
9
' gl_Position = u_ProjMatrix * a_Position;\n' +
10
' v_Color = a_Color;\n' +
11
'}\n';
...
23 function main() {
24
// Retrieve element
25
var canvas = document.getElementById('webgl');
26
// Retrieve the nearFar element
27
var nf = document.getElementById('nearFar');
...
52
// Get the storage location of u_ProjMatrix variable
53
var u_ProjMatrix = gl.getUniformLocation(gl.program,'u_ProjMatrix');
...
59
// Create the matrix to set the eye point and line of sight
60
var projMatrix = new Matrix4();
61
// Register the event handler to be called on key press
62
document.onkeydown = function(ev) { keydown(ev, gl, n, u_ProjMatrix,
➥projMatrix, nf); };
63
64
draw(gl, n, u_ProjMatrix, projMatrix, nf); // Draw triangles
65 }
...
116 // The distances to the near and far clipping plane

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117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140

var g_near = 0.0, g_far = 0.5;
function keydown(ev, gl, n, u_ProjMatrix, projMatrix, nf) {
switch(ev.keyCode) {
case 39: g_near += 0.01; break; // The right arrow key was pressed
case 37: g_near -= 0.01; break; // The left arrow key was pressed
case 38: g_far += 0.01; break; // The up arrow key was pressed
case 40: g_far -= 0.01; break; // The down arrow key was pressed
default: return; // Prevent the unnecessary drawing
}
draw(gl, n, u_ProjMatrix, projMatrix, nf);
}
function draw(gl, n, u_ProjMatrix, projMatrix, nf) {
// Set the viewing volume using a matrix
projMatrix.setOrtho(-1, 1, -1, 1, g_near, g_far);
// Set the projection matrix to u_ProjMatrix variable
gl.uniformMatrix4fv(u_ProjMatrix, false, projMatrix.elements);
gl.clear(gl.COLOR_BUFFER_BIT);

// Clear 

// Display the current near and far values
nf.innerHTML = 'near: ' + Math.round(g_near * 100)/100 + ', far: ' +
➥Math.round(g_far*100)/100;

141
142
gl.drawArrays(gl.TRIANGLES, 0, n);
143 }

// Draw the triangles

In a similar way to LookAtTrianglesWithKeys, keydown(), defined at line 118, is called
on key press, and draw() is called at the end of keydown() (line 127). The draw() function defined at line 130 sets the viewing volume, rewrites the near and far value on the
web page, and then draws the three triangles. The key point in this program is the draw()
function; however, before explaining the function, let’s quickly show how to rewrite
HTML elements using JavaScript.

Modifying an HTML Element Using JavaScript
The method of modifying an HTML element using JavaScript is similar to that of drawing
in a  with WebGL. That is, after retrieving the HTML element by using getElementById() and the id of the element, you write a message to the element in JavaScript.
In this sample program, you modify the following 
 element to show the message such
as “near: 0.0, far: 0.5”:
12

The near and far values are displayed here.

Specifying the Visible Range (Box Type)

247

This element is retrieved at line 27 in OrthoView.js using getElementById() as before.
Once you’ve retrieved the element, you need to specify the string ('nearFar') that was
bound to id at line 12 in the HTML file, as follows:
26
27

// Retrieve nearFar element
var nf = document.getElementById('nearFar');

Once you retrieve the 
 element into the variable nf (actually, nf is a JavaScript object),
you just need to change the content of this element. This is straightforward and uses the
innerHTML property of the object. For example, if you write:
nf.innerHTML = 'Good Morning, Marisuke-san!';

You will see the message “Good Morning, Marisuke-san!” on the web page. You can also
insert HTML tags in the message. For example, ‘Good Morning, Marisuke-san!’
will highlight “Marisuke.”
In OrthoView.js, you use the following equation to display the current near and far
values. These values are stored in the global variables g_near and g_far declared at line
117. When printing them, they are formatted using Math.round() as follows:
139
140

// Display the current near and far values
nf.innerHTML = 'near: ' + Math.round(g_near*100)/100 + ', far: ' +
➥Math.round(g_far*100)/100;

The Processing Flow of the Vertex Shader
As you can see with the following code, the processing flow in the vertex shader is almost
the same as that in LookAtRotatedTriangles.js except that the uniform variable name
(u_ProjMatrix) at line 6 was changed. This variable holds the matrix used to set the
viewing volume. So you just need to multiply the matrix (u_ProjMatrix) by the vertex
coordinates to set the viewing volume at line 9:
2 // Vertex shader program
3 var VSHADER_SOURCE =
...
6
'uniform mat4 u_ProjMatrix;\n' +
7
'varying vec4 v_Color;\n' +
8
'void main() {\n' +
9
' gl_Position = u_ProjMatrix * a_Position;\n' +
10
' v_Color = a_Color;\n' +
11
'}\n';

Line 62 registers the event handler for the arrow key press. Note that nf is passed as the
last argument to the handler to allow it to access the 
 element. The event handler use
the key press to determine the contents of the element in draw(), which is called in the
handler:

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61
62

// Register the event handler to be called on key press
document.onkeydown = function(ev) { keydown(ev, gl, n, u_ProjMatrix,
➥projMatrix, nf); };

The keydown() at line 121 identifies which arrow key is pressed and then modifies the
value of g_near and g_far before calling draw() at line 127. Line 117 defines g_near and
g_far, which are used by the setOrtho() method. These are defined as global variables
because they are used in both keydown() and draw():
116 // The distances to the near and far clipping plane
117 var g_near = 0.0, g_far = 0.5;
118 function keydown(ev, gl, n, u_ProjMatrix, projMatrix, nf) {
119
switch(ev.keyCode) {
120
case 39: g_near += 0.01; break; // The right arrow key was pressed
...
123
case 40: g_far -= 0.01; break; // The down arrow key was pressed
124
default: return; // Prevent the unnecessary drawing
125
}
126
127
draw(gl, n, u_ProjMatrix, projMatrix, nf);
128 }

Let’s examine the function draw(). The processing flow of draw(), defined at line 130, is
the same as in LookAtTrianglesWithKeys.js except for changing the message on the web
page at line 140:
130 function draw(gl, n, u_ProjMatrix, projMatrix, nf) {
131
// Set the viewing volume
132
projMatrix.setOrtho(-1.0, 1.0, -1.0, 1.0, g_near, g_far);
133
134
// Set the projection matrix to u_ProjMatrix variable
135
gl.uniformMatrix4fv(u_ProjMatrix, false, projMatrix.elements);
...
139
// Display the current near and far values
140
nf.innerHTML = 'near: ' + Math.round(g_near * 100)/100 + ', far: ' +
➥Math.round(g_far*100)/100;
141
142
gl.drawArrays(gl.TRIANGLES, 0, n); // Draw the triangles
143 }

Line 132 calculates the matrix for the viewing volume (projMatrix) and passes it to
u_ProjMatrix at line 135. Line 140 displays the current near and far value on the web
page. Finally, at line 142, the triangles are drawn.

Specifying the Visible Range (Box Type)

249

Changing Near or Far
When you run this program and increase the near value (right-arrow key), the display will
change, as shown in Figure 7.14.

Figure 7.14

Increase the near value using the right arrow key

By default, near is 0.0, so all three triangles are displayed. Next, when you increase near
using the right arrow key, the blue triangle (the front triangle) disappears because the
viewing volume moves past it, as shown in Figure 7.15. This result is shown as the middle
figure in Figure 7.14.
y

z
x

Figure 7.15

The blue triangle went outside the viewing volume

Again, if you continue to increase near by pressing the right arrow key, when near becomes
larger than 0.2, the near plane moves past the yellow triangle, so it is outside the viewing
volume and disappears. This leaves only the green triangle (the right figure in Figure 7.14).
At this point, if you use the left arrow key to decrease near so it becomes less than 0.2, the
yellow triangle becomes visible again. Alternatively, if you keep on increasing near, the
green triangle will also disappear, leaving the black canvas.

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As you can imagine, the behavior when you alter the far value is similar. As shown in
Figure 7.16, when far becomes less than 0.4, the back triangle (the green one) will disappear. Again, if you keep decreasing far, only the blue triangle will remain.

Figure 7.16

Decrease the far value using the down arrow key

This example should clarify the role of the viewing volume. Essentially, for any object you
want to display, you need to place it inside the viewing volume.

Restoring the Clipped Parts of the Triangles
(LookAtTrianglesWithKeys_ViewVolume.js)
In LookAtTrianglesWithKeys, when you kept pressing the arrow keys, part of the triangle
is clipped, as shown in Figure 7.17. From the previous discussion, it’s clear this is because
some part went outside the viewing volume. In this section, you will modify the sample
program to display the triangle correctly by setting the appropriate viewing volume.

Figure 7.17

A part of the triangle is clipped.
Specifying the Visible Range (Box Type)

251

As you can see from the figure, the far corner of the triangle from the eye point is clipped.
Obviously, the far clipping plane is too close to the eye point, so you need to move the far
clipping plane farther out than the current one. To achieve this, you can modify the arguments of the viewing volume so that left=–1.0, right=1.0, bottom=–1.0, top=1.0, near=0.0,
and far=2.0.
You will use two matrices in this program: the matrix that sets the viewing volume (the
orthographic projection matrix), and the matrix that sets the eye point and the line of
sight (view matrix). Because setOrtho() sets the viewing volume from the eye point, you
need to set the position of the eye point and then set the viewing volume. Consequently,
you will multiply the view matrix by the vertex coordinates to get the vertex coordinates, which are “viewed from the eye position” first, and then multiply the orthographic
projection matrix by the coordinates. You can calculate them as shown in Equation 7.3.
Equation 7.3
〈orthographic projection matrix 〉 × 〈view matrix 〉 × 〈vertex coordinates 〉
This can be implemented in the vertex shader, as shown in Listing 7.7.
Listing 7.7

LookAtTrianglesWithKeys_ViewVolume.js

1 // LookAtTrianglesWithKeys_ViewVolume.js
2 // Vertex shader program
3 var VSHADER_SOURCE =
4
'attribute vec4 a_Position;\n' +
5
'attribute vec4 a_Color;\n' +
6
'uniform mat4 u_ViewMatrix;\n' +
7
'uniform mat4 u_ProjMatrix;\n' +
8
'varying vec4 v_Color;\n' +
9
'void main() {\n' +
10
' gl_Position = u_ProjMatrix * u_ViewMatrix * a_Position;\n' +
11
' v_Color = a_Color;\n' +
12
'}\n';
...
24 function main() {
...
51
// Get the storage locations of u_ViewMatrix and u_ProjMatrix
52
varu_ViewMatrix = gl.getUniformLocation(gl.program,'u_ViewMatrix');
53
var u_ProjMatrix = gl.getUniformLocation(gl.program,'u_ProjMatrix');
...
59
// Create the matrix to specify the view matrix
60
var viewMatrix = new Matrix4();
61
// Register the event handler to be called on key press
62
document.onkeydown = function(ev) { keydown(ev, gl, n, u_ViewMatrix,
➥viewMatrix); };

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63
64
65
66
67
68
69
70 }

// Create the matrix to specify the viewing volume and pass it to u_ProjMatrix
var projMatrix = new Matrix4();
projMatrix.setOrtho(-1.0, 1.0, -1.0, 1.0, 0.0, 2.0);
gl.uniformMatrix4fv(u_ProjMatrix, false, projMatrix.elements);
draw(gl, n, u_ViewMatrix, viewMatrix); // Draw the triangles

Line 66 calculates the orthographic projection matrix (projMatrix) by modifying far from
1.0 to 2.0. The result matrix is passed to u_ProjMatrix in the vertex shader at line 67.
A uniform variable is used because the elements in the matrix are uniform for all vertex
coordinates. If you run this sample program and move the eye point as before, you can
see that the triangle no longer gets clipped (see Figure 7.18).

Figure 7.18

LookAtTrianglesWithKeys_ViewVolume

Experimenting with the Sample Program
As we explained in the section “Specify the Viewing Volume,” if the aspect ratio of
 is different from that of the near clipping plane, distorted objects are displayed.
Let’s explore this. First, in OrthoView_halfSize (based on Listing 7.7), you reduce the
current size of the near clipping plane to half while keeping its aspect ratio:
projMatrix.setOrtho(-0.5, 0.5, -0.5, 0.5, 0, 0.5);

The result is shown on the left of Figure 7.19. As you can see, the triangles appear twice
as large as those of the previous sample because the size of  is the same as before.
Note that the parts of the triangles outside the near clipping plane are clipped.

Specifying the Visible Range (Box Type)

253

Figure 7.19

Modify the size of the near clipping plane

In OrthoView_halfWidth, you reduce only the width of the near clipping plane by changing the first two arguments in setOrtho() as follows:
projMatrix.setOrtho(-0.3, 0.3, -1.0, 1.0, 0.0, 0.5);

You can see the results on the right side of Figure 7.19. This is because the near clipping
plane is horizontally reduced and then horizontally extended (and thus distorted) to fit
the square-shaped  when the plane is displayed.

Specifying the Visible Range Using a Quadrangular
Pyramid
Figure 7.20 shows a tree-lined road scene. In this picture, all the trees on the left and right
sides are approximately of the same height, but the farther back they are, the smaller
they look. Equally, the building in the distance appears smaller than the trees that are
closer to the viewer, even though the building is actually taller than the trees. This effect
of distant objects looking smaller gives the feeling of depth. Although our eyes perceive
reality in this way, it’s interesting to notice that children’s drawings rarely show this kind
of perspective.

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Figure 7.20

Tree-lined road

In the case of the box-shaped viewing volume explained in the previous section, identically sized triangles are drawn the same size, regardless of their distance from the eye
point. To overcome this constraint, you can use the quadrangular pyramid viewing
volume, which allows you to give this sense of depth, as seen in Figure 7.20.
Here you construct the sample program PerspectiveView, which sets a quadrangular
pyramid viewing volume that points along the negative z-axis from the eye point set at
(0, 0, 5). Figure 7.21 shows a screen shot of PerspectiveView and the location of each
triangle.

y
1.0

-0.75
0.75

z

-1.0

-4.0

x
-2.0
0.0

Figure 7.21

PerspectiveView; location of each triangle
Specifying the Visible Range Using a Quadrangular Pyramid

255

As can be seen from the figure on the right, three identically sized triangles are positioned
on the right and left sides along the coordinate’s axes, in a way similar to the tree-lined
road. By using a quadrangular pyramid viewing volume, WebGL can automatically display
remote objects as if they are smaller, thus achieving the sense of depth. This is shown in
the left side of the figure.
To really notice the change in size, as in the real world, the objects need to be located
at a substantial distance. For example, when looking at the box, to actually make the
background area looks smaller than the foreground area, this box needs to have considerable depth. So this time, you will use a slightly more distant position (0, 0, 0.5) than the
default value (0, 0, 0) for the eye point.

Setting the Quadrangular Pyramid Viewing Volume
The quadrangular pyramid viewing volume is shaped as shown in Figure 7.22. Just like
the box-shaped configuration, the viewing volume is set at the eye point along the line of
sight, and objects located between the far and near clipping planes are displayed. Objects
positioned outside the viewing volume are not shown, while those straddling the boundary will only have parts located inside the viewing volume visible.
far clipping plane
visible
invisible
near clipping plane

eye point
y

up
vector

eye point

invisible

fo v

z

line of sight

near
far

Figure 7.22

aspect
(aspect of near clipping plane)

Quadrangular pyramid viewing volume

Regardless of whether it is a quadrangular pyramid or a box, you set the viewing volume
using matrices, but the arguments differ. The Matrix4’s method setPerspective() is used
to configure the quadrangular pyramid viewing volume.

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Matrix4.setPerspective(fov, aspect, near, far)
Calculate the matrix (the perspective projection matrix) that defines the viewing volume
specified by its arguments, and store it in Matrix4. However, the near value must be less
than the far value.
Parameters

Return value

fov

Specifies field of view, angle formed by the top and bottom
planes. It must be greater than 0.

aspect

Specifies the aspect ratio of the near plane (width/height).

near, far

Specify the distances to the near and far clipping planes along the
line of sight (near > 0 and far > 0).

None

The matrix that sets the quadrangular pyramid viewing volume is called the perspective
projection matrix.
Note that the specification for the near plane is different from that of the box type with
the second argument, aspect, representing the near plane aspect ratio. For example, if we
set the height to 100 and the width to 200, the aspect ratio is 0.5.
The positioning of the triangles with regard to the viewing volume we are using is illustrated in Figure 7.23. It is specified by near=1.0, far=100, aspect=1.0 (the same aspect ratio
as the canvas), and fov=30.0.

y
1.0

-0.75

eye point

0.75

x

-1.0

(0, 0, 4)

z

(0, 0, 5)

far=100
near = 1

Figure 7.23
volume

The positions of the triangles with respect to the quadrangular pyramid viewing
Specifying the Visible Range Using a Quadrangular Pyramid

257

The basic processing flow is similar to that of LookAtTrianglesWithKeys_ViewVolume.js in
the previous section. So let’s take a look at the sample program.

Sample Program (PerspectiveView.js)
The sample program is detailed in Listing 7.8.
Listing 7.8

PerspectiveView.js

1 // PerspectiveView.js
2 // Vertex shader program
3 var VSHADER_SOURCE =
4
'attribute vec4 a_Position;\n' +
5
'attribute vec4 a_Color;\n' +
6
'uniform mat4 u_ViewMatrix;\n' +
7
'uniform mat4 u_ProjMatrix;\n' +
8
'varying vec4 v_Color;\n' +
9
'void main() {\n' +
10
' gl_Position = u_ProjMatrix * u_ViewMatrix * a_Position;\n' +
11
' v_Color = a_Color;\n' +
12
'}\n';
...
24 function main() {
...
41
// Set the vertex coordinates and color (blue triangle is in front)
42
var n = initVertexBuffers(gl);
...
51
// Get the storage locations of u_ViewMatrix and u_ProjMatrix
52
varu_ViewMatrix = gl.getUniformLocation(gl.program,'u_ViewMatrix');
53
var u_ProjMatrix = gl.getUniformLocation(gl.program,'u_ProjMatrix');
...
59
var viewMatrix = new Matrix4(); // The view matrix
60
var projMatrix = new Matrix4(); // The projection matrix
61
62
// Calculate the view and projection matrix
63
viewMatrix.setLookAt(0, 0, 5, 0, 0, -100, 0, 1, 0);
64
projMatrix.setPerspective(30, canvas.width/canvas.height, 1, 100);
65
// Pass The view matrix and projection matrix to u_ViewMatrix and u_ProjMatrix
66
gl.uniformMatrix4fv(u_ViewMatrix, false, viewMatrix.elements);
67
gl.uniformMatrix4fv(u_ProjMatrix, false, projMatrix.elements);
...
72
// Draw the rectangles
73
gl.drawArrays(gl.TRIANGLES, 0, n);
74 }
75

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76 function initVertexBuffers(gl) {
77
var verticesColors = new Float32Array([
78
// Three triangles on the right side
79
0.75, 1.0, -4.0, 0.4, 1.0, 0.4, // The
80
0.25, -1.0, -4.0, 0.4, 1.0, 0.4,
81
1.25, -1.0, -4.0, 1.0, 0.4, 0.4,
82
83
0.75, 1.0, -2.0, 1.0, 1.0, 0.4, // The
84
0.25, -1.0, -2.0, 1.0, 1.0, 0.4,
85
1.25, -1.0, -2.0, 1.0, 0.4, 0.4,
86
87
0.75, 1.0, 0.0, 0.4, 0.4, 1.0, // The
88
0.25, -1.0, 0.0, 0.4, 0.4, 1.0,
89
1.25, -1.0, 0.0, 1.0, 0.4, 0.4,
90
91
// Three triangles on the left side
92
-0.75, 1.0, -4.0, 0.4, 1.0, 0.4, // The
93
-1.25, -1.0, -4.0, 0.4, 1.0, 0.4,
94
-0.25, -1.0, -4.0, 1.0, 0.4, 0.4,
95
96
-0.75, 1.0, -2.0, 1.0, 1.0, 0.4, // The
97
-1.25, -1.0, -2.0, 1.0, 1.0, 0.4,
98
-0.25, -1.0, -2.0, 1.0, 0.4, 0.4,
99
100
-0.75, 1.0, 0.0, 0.4, 0.4, 1.0, // The
101
-1.25, -1.0, 0.0, 0.4, 0.4, 1.0,
102
-0.25, -1.0, 0.0, 1.0, 0.4, 0.4,
103
]);
104
var n = 18; // Three vertices per triangle
...
138
return n;
139 }

green triangle in back

yellow triangle in middle

blue triangle in front

green triangle in back

yellow triangle in middle

blue triangle in front

* 6

The vertex and fragment shaders are completely identical (including the names of the
variables) to the ones used in LookAtTriangles_ViewVolume.js.
The processing flow of main() in JavaScript is also similar. Calling initVertexBuffers()
at line 42 writes the vertex coordinates and colors of the six triangles to be displayed into
the buffer object. In initVertexBuffers(), the vertex coordinates and colors for the six
triangles are specified: three triangles positioned on the right side from line 79 and three
triangles positioned on the left side from line 92. As a result, the number of vertices to be
drawn at line 104 is changed to 18 (3×6=18, to handle six triangles).
At lines 52 and 53 in main(), the locations of the uniform variables that store the view
matrix and perspective projection matrix are retrieved. Then at line 59 and 60, the variables used to hold the matrices are created.
Specifying the Visible Range Using a Quadrangular Pyramid

259

At line 63, the view matrix is calculated, with the eye point set at (0, 0, 5), the line of
sight set along the z-axis in the negative direction, and the up direction set along the
y-axis in the positive direction. Finally at line 64, the projection matrix is set up using a
quadrangular pyramid viewing volume:
64

projMatrix.setPerspective(30, canvas.width/canvas.height, 1, 100);

The second argument aspect (the horizontal to vertical ratio of the near plane) is derived
from the  width and height (width and height property), so any modification of
the  aspect ratio doesn’t lead to distortion of the objects displayed.
Next, as the view and perspective projection matrices are available, you pass them to the
appropriate uniform variables at lines 66 and 67. Finally, you draw the triangles at line 73,
and upon execution you get a result including perspective similar to that shown in Figure
7.20.
Finally, one aspect touched on earlier but not fully explained is why matrices are used to
set the viewing volume. Without using mathematics, let’s explore that a little.

The Role of the Projection Matrix
Let’s start by examining the perspective projection matrix. Looking at the screen shot of
PerspectiveView in Figure 7.24, you can see that, after applying the projection matrix, the
objects in the distance are altered in two ways.

Figure 7.24

PerspectiveView

First, the farther away the triangles are, the smaller they appear. Second, the triangles are
parallel shifted so they look as if they are positioned inward toward the line of sight. In
comparison to the identically sized triangles that are laid out as shown on the left side
of Figure 7.25, the following two transformations have been applied: (1) triangles farther

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from the viewer are scaled down (transformed) in proportion to the distance from the
viewer, and (2) the triangles are then transformed to be shifted toward the line of sight, as
illustrated on the right side of Figure 7.25. These two transformations, shown on the right
side of Figure 7.25, enable the effect you see in the photograph scene shown in Figure
7.20.
y

point at
infinity

y

z

x
z

Figure 7.25

x

Conceptual rendering of the perspective projection transformation

This means that the specification of the viewing volume can be represented as a combination of transformations, such as the scaling or translation of geometric shapes and objects,
in accordance with the shape of the viewing volume. The Matrix4 object’s method
setPerspective() automatically calculates this transformation matrix from the arguments
of the specified viewing volume. The elements of the matrix are discussed in Appendix C,
“Projection Matrices.” If you are interested in the mathematical explanation of the coordinate transform related to the viewing volume, please refer to the book Computer Graphics.
To put it another way, the transformation associated with the perspective projection transforms the quadrangular pyramid viewing volume into a box-shaped viewing volume (right
part of Figure 7.25).
Note that the orthographic projection matrix does not perform all the work needed for
this transformation to generate the required optical effect. Rather, it performs the preliminary preparation that is required by the post vertex shader processing—where the actual
processing is done. If you are interested in this, please refer to Appendix D, “WebGL/
OpenGL: Left or Right Handed?”
The projection matrix, combined with the model matrix and the view matrix, is able
to handle all the necessary geometric transformations (translation, rotation, scaling) for
achieving the different optical effects. The following section will explore how to combine
these matrices to do that using a simple example.

Specifying the Visible Range Using a Quadrangular Pyramid

261

Using All the Matrices (Model Matrix, View Matrix, and Projection
Matrix)
One of the issues with PerspectiveView.js is the amount of code needed to set up the
vertex coordinates and the color data. Because we only have to deal with six triangles in
this case, it’s still manageable, but it could get messy if the number of triangles increased.
Fortunately, there is an effective drawing technique to handle this problem.
If you take a close look at the triangles, you will notice that the configuration is identical
to that in Figure 7.26, where the dashed triangles are shifted along the x-axis in the positive (0.75) and negative (–0.75) directions, respectively.

y

-0.75
0.75

x

z
Figure 7.26

Drawing after translation

Taking advantage of this, it is possible to draw the triangles in PerspectiveView in the
following way:
1. Prepare the vertex coordinates data of the three triangles that are laid out centered
along the z-axis.
2. Translate the original triangles by 0.75 units along the x-axis, and draw them.
3. Translate the original triangles by –0.75 units along the x-axis, and draw them.
Now let’s try to use this approach in some sample code (PerspectiveView_mvp).
In the original PerspectiveView program the projection and view matrices were used to
specify the viewer’s viewpoint and viewing volume and PerspectiveView_mvp, the model
matrix, was used to perform the translation of the triangles.
At this point, it’s worthwhile to review the actions these matrices perform. To do that,
let’s refer to LookAtTriangles, which you wrote earlier to allow the viewer to look at a
rotated triangle from a specific location. At that time, you used this expression, which is
identical to Equation 7.1:

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〈view matrix 〉 × 〈 model matrix 〉 × 〈vertex coordinates 〉
Building on that, in LookAtTriangles_ViewVolume, which correctly displays the clipped
triangle, you used the following expression, which, when you use projection matrix
to include either orthographic projection or perspective projection, is identical to
Equation 7.3:
〈 projection matrix 〉 × 〈view matrix 〉 × 〈vertex coordinates 〉
You can infer the following from these two expressions:
Equation 7.4
〈 projection matrix 〉 × 〈view matrix 〉 × 〈 model matrix 〉 × 〈vertex coordinates 〉
This expression shows that, in WebGL, you can calculate the final vertex coordinates
by using three types of matrices: the model matrix, the view matrix, and the projection
matrix.
This can be understood by considering that Equation 7.1 is identical to Equation 7.4, in
which the projection matrix becomes the identity matrix, and Equation 7.3 is identical
to Equation 7.4, whose model matrix is turned into the identity matrix. As explained in
Chapter 4, the identity matrix behaves for matrix multiplication like the scalar 1 does with
scalar multiplication. Multiplying by the identity matrix has no effect on the other matrix.
So let’s construct the sample program using Equation 7.4.

Sample Program (PerspectiveView_mvp.js)
PerspectiveView_mvp.js is shown in Listing 7.9. The basic processing flow is similar to
that of PerspectiveView.js. The only difference is the modification of the calculation in

the vertex shader (line 11) to implement Equation 7.4, and the passing of the additional
matrix (u_ModelMatrix) used for the calculation.
Listing 7.9

PerspectiveView_mvp.js

1 // PerspectiveView_mvp.js
2 // Vertex shader program
3 var VSHADER_SOURCE =
4
'attribute vec4 a_Position;\n' +
5
'attribute vec4 a_Color;\n' +
6
'uniform mat4 u_ModelMatrix;\n' +
7
'uniform mat4 u_ViewMatrix;\n' +
8
'uniform mat4 u_ProjMatrix;\n' +
9
'varying vec4 v_Color;\n' +
10
'void main() {\n' +
11
' gl_Position = u_ProjMatrix * u_ViewMatrix * u_ModelMatrix * a_Position;\n' +
12
' v_Color = a_Color;\n' +

Specifying the Visible Range Using a Quadrangular Pyramid

263

13
25
42
43
52
53
54
55

'}\n';
...
function main() {
...
// Set the vertex coordinates and color (blue triangle is in front)
var n = initVertexBuffers(gl);
...
// Get the storage locations of u_ModelMatrix, u_ViewMatrix, and u_ProjMatrix.
var u_ModelMatrix = gl.getUniformLocation(gl.program, 'u_ModelMatrix');
var u_ViewMatrix = gl.getUniformLocation(gl.program,'u_ViewMatrix');
var u_ProjMatrix = gl.getUniformLocation(gl.program,'u_ProjMatrix');
...
var modelMatrix = new Matrix4(); // Model matrix
var viewMatrix = new Matrix4(); // View matrix
var projMatrix = new Matrix4(); // Projection matrix

61
62
63
64
65
// Calculate the model matrix, view matrix, and projection matrix
66
modelMatrix.setTranslate(0.75, 0, 0); // Translate 0.75 units
67
viewMatrix.setLookAt(0, 0, 5, 0, 0, -100, 0, 1, 0);
68
projMatrix.setPerspective(30, canvas.width/canvas.height, 1, 100);
69
// Pass the model, view, and projection matrix to uniform variables.
70
gl.uniformMatrix4fv(u_ModelMatrix, false, modelMatrix.elements);
71
gl.uniformMatrix4fv(u_ViewMatrix, false, viewMatrix.elements);
72
gl.uniformMatrix4fv(u_ProjMatrix, false, projMatrix.elements);
73
74
gl.clear(gl.COLOR_BUFFER_BIT);// clear 
75
76
gl.drawArrays(gl.TRIANGLES, 0, n); // Draw triangles on right
77
78
// Prepare the model matrix for another pair of triangles
79
modelMatrix.setTranslate(-0.75, 0, 0); // Translate -0.75
80
// Modify only the model matrix
81
gl.uniformMatrix4fv(u_ModelMatrix, false, modelMatrix.elements);
82
83
gl.drawArrays(gl.TRIANGLES, 0, n);// Draw triangles on left
84 }
85
86 function initVertexBuffers(gl) {
87
var verticesColors = new Float32Array([
88
// Vertex coordinates and color
89
0.0, 1.0, -4.0, 0.4, 1.0, 0.4, // The back green triangle
90
-0.5, -1.0, -4.0, 0.4, 1.0, 0.4,
91
92
93

0.5, -1.0,

-4.0, 1.0,

0.4, 0.4,

0.0,

-2.0, 1.0,

1.0, 0.4, // The middle yellow triangle

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1.0,

Toward the 3D World

94
95
96
97
98
99
100

-0.5, -1.0,
0.5, -1.0,

-2.0, 1.0,
-2.0, 1.0,

1.0, 0.4,
0.4, 0.4,

0.0, 1.0,
-0.5, -1.0,
0.5, -1.0,
]);
...
135
return n;
136 }

0.0, 0.4,
0.0, 0.4,
0.0, 1.0,

0.4, 1.0,
0.4, 1.0,
0.4, 0.4,

// The front blue triangle

This time, you need to pass the model matrix to the vertex shader, so u_ModelMatrix is
added at line 6. The matrix is used at line 11, which implements Equation 7.5:
11

'

gl_Position = u_ProjMatrix * u_ViewMatrix * u_ModelMatrix * a_Position;\n' +

Next, main() in JavaScript calls initVertexBuffers() at line 43. In this function, the
vertex coordinates of the triangles to be passed to the buffer object are defined (line 87).
This time, you are handling the vertex coordinates of three triangles centered along the
z-axis instead of the six triangles used in PerspectiveView.js. As mentioned before, this is
because you will use the three triangles in conjunction with a translation.
At line 53, the storage location of u_ModelMatrix in the vertex shader is obtained. At
line 61, the arguments for the matrix (modelMatrix) passed to the uniform variable are
prepared, and at line 66, the matrix is calculated. First, this matrix will translate the triangles by 0.75 units along the x-axis:
65
66
...
70
...
76

// Calculate the view matrix and the projection matrix
modelMatrix.setTranslate(0.75, 0, 0); // Translate 0.75
gl.uniformMatrix4fv(u_ModelMatrix, false, modelMatrix.elements);
gl.drawArrays(gl.TRIANGLES, 0, n);

// Draw a triangle

The matrix calculations, apart from the model matrix at line 66, are the same as in
PerspectiveView.js. The model matrix is passed to u_ModelMatrix at line 70 and used to
draw the right side row of triangles (line 76).
In a similar manner, the row of triangles for the left side is translated by –0.75 units along
the x-axis, and then the model matrix is calculated again at line 79. Because the view
matrix and projection matrix make use of this model matrix, you only need to assign
the model matrix to the uniform variable once (line 81). Once the matrix is set up, you
perform the draw operation at line 83 with gl.drawArrays():
78
79
80

// Prepare the model matrix for another pair of triangles
modelMatrix.setTranslate(-0.75, 0, 0); // Translate -0.75
// Modify only the model matrix

Specifying the Visible Range Using a Quadrangular Pyramid

265

81
82
83

gl.uniformMatrix4fv(u_ModelMatrix, false, modelMatrix.elements);
gl.drawArrays(gl.TRIANGLES, 0, n);

// Draw triangles on left

As you have seen, this approach allows you to draw two sets of triangles from a single set
of triangle data, which reduces the number of vertices needed but increases the number
of calls to gl.drawArrays(). The choice of which approach to use for better performance
depends on the application and the WebGL implementation.

Experimenting with the Sample Program
In PerspectiveView_mvp, you calculated 〈 projection matrix 〉 × 〈view matrix 〉 × 〈 model matrix 〉
directly inside the vertex shader. This calculation of is the same for all the vertices, so
there is no need to recalculate it inside the shader for each vertex. It can be computed in
advance inside the JavaScript code, as it was in LookAtRotatedTriangles_mvMatrix earlier
in the chapter, allowing a single matrix to be passed to the vertex shader. This matrix is
called the model view projection matrix, and the name of the variable that passes it is
u_MvpMatrix. The sample program used to show this is ProjectiveView_mvpMatrix, in
which the vertex shader is modified as shown next and, as you can see, is significantly
simpler:
1 // PerspectiveView_mvpMatrix.js
2 // Vertex shader program
3 var VSHADER_SOURCE =
4
'attribute vec4 a_Position;\n' +
5
'attribute vec4 a_Color;\n' +
6
'uniform mat4 u_MvpMatrix;\n' +
7
'varying vec4 v_Color;\n' +
8
'void main() {\n' +
9
' gl_Position = u_MvpMatrix * a_Position;\n' +
10
' v_Color = a_Color;\n' +
11
'}\n';

In JavaScript, main(), the storage location of u_ModelMatrix is retrieved at line 51, and
then the matrix to be stored in the uniform variable is calculated at line 57:
50
51
57
58

266

// Get the storage location of u_MvpMatrix
var u_MvpMatrix = gl.getUniformLocation(gl.program, 'u_MvpMatrix');
...
var modelMatrix = new Matrix4(); // The model matrix
var viewMatrix = new Matrix4(); // The view matrix

CHAPTER 7

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59
60
61
62
63
64
65
66
67
68
69

var projMatrix = new Matrix4();
var mvpMatrix = new Matrix4();

// The projection matrix
// The model view projection matrix

73
74
75
76
77
78
79
80
81
82
83 }

// Calculate the model, view, and projection matrices
modelMatrix.setTranslate(0.75, 0, 0);
viewMatrix.setLookAt(0, 0, 5, 0, 0, -100, 0, 1, 0);
projMatrix.setPerspective(30, canvas.width/canvas.height, 1, 100);
// Calculate the model view projection matrix
mvpMatrix.set(projMatrix).multiply(viewMatrix).multiply(modelMatrxi);
// Pass the model view projection matrix to u_MvpMatrix
gl.uniformMatrix4fv(u_MvpMatrix, false, mvpMatrix.elements);
...
gl.drawArrays(gl.TRIANGLES, 0, n); // Draw a rectangle
// Prepare the model matrix for another pair of triangles
modelMatrix.setTranslate(-0.75, 0, 0);
// Calculate the model view projection matrix
mvpMatrix.set(projMatrix).multiply(viewMatrix).multiply(modelMatrxi);
// Pass the model view projection matrix to u_MvpMatrix
gl.uniformMatrix4fv(u_MvpMatrix, false, mvpMatrix.elements);
gl.drawArrays(gl.TRIANGLES, 0, n); // Draw a rectangle

The critical calculation is carried out at line 67. The projection matrix (projMatrix) is
assigned to mvpMatrix. Then the view matrix (viewMatrix) is multiplied by the model
matrix (modelMatrix), and the result is written back to mvpMatrix, using the set version
of the method. This is in turn assigned to u_MvpMatrix at line 69, and the triangles on
the right side are drawn at line 73. Similarly, the calculation of the model view projection matrix for the triangles on the left side is performed at line 78. It is then passed to
u_MvpMatrix at line 80, and the triangles are drawn at line 82.
With this information, you are now able to write code that moves the eye point, sets the
viewing volume, and allows you to view three-dimensional objects from various angles.
Additionally, you have learned how to deal with clipping that resulted in partially missing
objects. However, one potential problem remains. As you move the eye point to a different location, it’s possible for objects in the foreground to be hidden by objects in the
background. Let’s look at how this problem comes about.

Correctly Handling Foreground and Background
Objects
In the real world, if you place two boxes on a desk as shown in Figure 7.27, the foreground box partially hides the background one.

Correctly Handling Foreground and Background Objects

267

Figure 7.27

The front object partially hides the back object

Looking at the sample programs constructed so far, such as the screen shot of
PerspectiveView (refer to Figure 7.21), the green triangle located at the back is partially

hidden by the yellow and blue triangles. It looks as if WebGL, being designed for displaying 3D objects, has naturally figured out the correct order.
However, that is unfortunately not the case. By default, WebGL, to accelerate the drawing
process, draws objects in the order of the vertices specified inside the buffer object. Up
until now, you have always arranged the order of the vertices so that the objects located
in the background are drawn first, thus resulting in a natural rendering.
For example, in PerspectiveView_mvpMatrix.js, you specified the coordinates and color
of the triangles in the following order. Note the z coordinates (bold font):
var verticesColors = new Float32Array([
// vertex coordinates and color
0.0, 1.0, -4.0, 0.4, 1.0, 0.4, // The green one at the back
-0.5, -1.0, -4.0, 0.4, 1.0, 0.4,
0.5, -1.0, -4.0, 1.0, 0.4, 0.4,
0.0, 1.0, -2.0, 1.0, 1.0, 0.4, // The yellow one in the middle
-0.5, -1.0, -2.0, 1.0, 1.0, 0.4,
0.5, -1.0, -2.0, 1.0, 0.4, 0.4,
0.0, 1.0, 0.0, 0.4, 0.4, 1.0, // The blue one in the front
-0.5, -1.0, 0.0, 0.4, 0.4, 1.0,
0.5, -1.0, 0.0, 1.0, 0.4, 0.4,
]);

WebGL draws the triangles in the order z in which you specified the vertices (that is,
the green triangle [back], then the yellow triangle [middle], and finally the blue triangle

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[front]). This ensures that objects closer to the eye point cover those farther away, as seen
in Figure 7.13.
To verify this, let’s modify the order in which the triangles are specified by first drawing
the blue triangle in the front, then the yellow triangle in the middle, and finally the green
triangle at the back:
var verticesColors = new Float32Array([
// vertex coordinates and color
0.0, 1.0,
0.0, 0.4, 0.4, 1.0, // The blue one in the front
-0.5, -1.0,
0.0, 0.4, 0.4, 1.0,
0.5, -1.0,
0.0, 1.0, 0.4, 0.4
0.0, 1.0,
-0.5, -1.0,
0.5, -1.0,

-2.0,
-2.0,
-2.0,

1.0,
1.0,
1.0,

1.0,
1.0,
0.4,

0.4, // The yellow one in the middle
0.4,
0.4,

0.0, 1.0,
-0.5, -1.0,
0.5, -1.0,
]);

-4.0,
-4.0,
-4.0,

0.4,
0.4,
1.0,

1.0,
1.0,
0.4,

0.4, // The green one at the back
0.4,
0.4,

When you run this, you’ll see the green triangle, which is supposed to be located at the
back, has been drawn at the front (see Figure 7.28).

Figure 7.28

The green triangle in the back is displayed at the front

Drawing objects in the specified order, the default behavior in WebGL, can be quite
efficient when the sequence can be determined beforehand and the scene doesn’t

Correctly Handling Foreground and Background Objects

269

subsequently change. However, when you examine the object from various directions by
moving the eye point, it is impossible to decide the drawing order in advance.

Hidden Surface Removal
To cope with this problem, WebGL provides a hidden surface removal function. This
function eliminates surfaces hidden behind foreground objects, allowing you to draw the
scene so that the objects in the back are properly hidden by those in front, regardless of
the specified vertex order. This function is already embedded in WebGL and simply needs
to be enabled.
Enabling hidden surface removal and preparing WebGL to use it requires the following
two steps:
1. Enabling the hidden surface removal function
gl.enable(gl.DEPTH_TEST);

2. Clearing the depth buffer used for the hidden surface removal before drawing
gl.clear(gl.DEPTH_BUFFER_BIT);

The function gl.enable(), used in step 1, actually enables various functions in WebGL.

gl.enable(cap)
Enable the function specified by cap (capability).
Parameters

cap

Specifies the function to be
enabled.

Return value

None

Errors:

INVALID_ENUM

gl.DEPTH_TEST2

Hidden surface removal

gl.BLEND

Blending (see Chapter 9,
“Hierarchical Objects”)

gl.POLYGON_OFFSET_FILL

Polygon offset (see the next
section), and so on3

None of the acceptable values is specified in cap

2

A “DEPTH_TEST” in the hidden surface removal function might sound strange, but actually its name
comes from the fact that it decides which objects to draw in the foreground by verifying (TEST) the
depth (DEPTH) of each object.

3

Although not covered in this book, you can also specify gl.CULL_FACE, gl.DITHER, gl.SAMPLE_
ALPHA_TO_COVERAGE, gl.SAMPLE_COVERAGE, gl.SCISSOR_TEST, and gl.STENCIL_TEST. See the
book OpenGL Programming Guide for more information on these.

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The depth buffer cleared in the gl.clear() statement (step 2) is a buffer used internally
to remove hidden surfaces. While WebGL draws objects and geometric shapes in the color
buffer displayed on the , hidden surface removal requires the depth (from the
eye point) for each geometrical shape and object. The depth buffer holds this information
(see Figure 7.29). The depth direction is the same as the z-axis direction, so it is sometimes
called the z-buffer.
Hidden Surface Removal

JavaScript

Fragment
Shader

Vertex Shader

y

function m a in() {
var gl=getW ebG L…
…
initS haders( … ) ;
…
}

Color Buffer

Depth
Test
Depth Buffer

x

WebGL System

Figure 7.29

Depth buffer used in hidden surface removal

Because the depth buffer is used whenever a drawing command is issued, it must be
cleared before any drawing operation; otherwise, you will see incorrect results. You specify
the depth buffer using gl.DEPTH_BUFFER_BIT and proceed as follows to clear it:
gl.clear(gl.DEPTH_BUFFER_BIT);

Up until now, you only cleared the color buffer. Because you now need to also clear the
depth buffer, you can clear both buffers simultaneously by taking the bitwise or (|) of
gl.COLOR_BUFFER_BIT (which represents the color buffer) and gl.DEPTH_BUFFER_BIT
(which represents the depth buffer) and specifying it as an argument to gl.clear():
gl.clear(gl.COLOR_BUFFER_BIT | gl.DEPTH_BUFFER_BIT);

You can use the bitwise or operation this way whenever you need to clear both buffers at
the same time.
To disable the function you enabled with gl.enable(), you use gl.disable().

gl.disable(cap)
Disable the function specified by cap (capability).
Parameters

cap

Return value

None

Errors

INVALID_ENUM

Same as gl.enable().

None of the acceptable values is specified in cap

Correctly Handling Foreground and Background Objects

271

Sample Program (DepthBuffer.js)
Let’s add the hidden surface removal methods from (1) and (2) to PerspectiveView_
mvpMatrix.js and change the name to DepthBuffer.js. Note that the order of the vertex
coordinates specified inside the buffer object is not changed, so you will draw from
front to back the blue, yellow, and green triangles. The result is identical to that of the
PerspectiveView_mvpMatrix. We detail the program in Listing 7.10.
Listing 7.10

DepthBuffer.js

1 // DepthBuffer.js
...
23 function main() {
...
41
var n = initVertexBuffers(gl);
...
47
// Specify the color for clearing 
48
gl.clearColor(0, 0, 0, 1);
49
// Enable the hidden surface removal function
50
gl.enable(gl.DEPTH_TEST);
73
74
75
76

// Clear the color and depth buffer
gl.clear(gl.COLOR_BUFFER_BIT | gl.DEPTH_BUFFER_BIT);

gl.drawArrays(gl.TRIANGLES, 0, n); // Draw triangles
...
85
gl.drawArrays(gl.TRIANGLES, 0, n); // Draw triangles
86 }
87
88 function initVertexBuffers(gl) {
89
var verticesColors = new Float32Array([
90
// Vertex coordinates and color
91
0.0, 1.0,
0.0, 0.4, 0.4, 1.0, // The blue triangle in front
92
-0.5, -1.0,
0.0, 0.4, 0.4, 1.0,
93
0.5, -1.0,
0.0, 1.0, 0.4, 0.4,
94
95
0.0, 1.0, -2.0, 1.0, 1.0, 0.4, // The yellow triangle in middle
96
-0.5, -1.0, -2.0, 1.0, 1.0, 0.4,
97
0.5, -1.0, -2.0, 1.0, 0.4, 0.4,
98
99
0.0, 1.0, -4.0, 0.4, 1.0, 0.4, // The green triangle in back
100
-0.5, -1.0, -4.0, 0.4, 1.0, 0.4,
101
0.5, -1.0, -4.0, 1.0, 0.4, 0.4,
102
]);
103
var n = 9;
...

272

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137
return n;
138 }

If you run DepthBuffer, you can see that hidden face removal is performed and that
objects placed at the back are hidden by objects located at the front. This demonstrates
that the hidden surface removal function can eliminate the hidden surfaces regardless of
the position of the eye point. Equally, this also shows that in anything but a trivial 3D
scene, you will always need to enable hidden surface removal and systematically clear the
depth buffer before any drawing operation.
You should note that hidden surface removal requires you to correctly set up the viewing
volume. If you fail to do this (use WebGL in its default configuration), you are likely
to see incorrect results. You can specify either a box or a quadrangular pyramid for the
viewing volume.

Z Fighting
Hidden surface removal is a sophisticated and powerful feature of WebGL that correctly
handles most of the cases where surfaces need to be removed. However, it fails when two
geometrical shapes or objects are located at extremely close positions and results in the
display looking a little unnatural. This phenomenon is known as Z fighting and is illustrated in Figure 7.30. Here, we draw two triangles sharing the same z coordinate.

Figure 7.30

Visual artifact generated by Z fighting (the left side)

The Z fighting occurs because of the limited precision of the depth buffer and means the
system is unable to asses which object is in front and which is behind. Technically, when
handling 3D models, you could avoid this by paying thorough attention to the z coordinates’ values at the model creation stage; however, implementing this workaround would
prove to be unrealistic when dealing with the animation of several objects.
Correctly Handling Foreground and Background Objects

273

To help resolve this problem, WebGL provides a feature known as the polygon offset.
This works by automatically adding an offset to the z coordinate, whose value is a function of each object’s inclination with respect to the viewer’s line of sight. You only need
to add two lines of code to enable this function.
1. Enabling the polygon offset function:
gl.enable(gl.POLYGON_OFFSET_FILL);

2. Specifying the parameter used to calculate the offset (before drawing):
gl.polygonOffset(1.0, 1.0);

The same method that enabled the hidden surface removal function is used, but with a
different parameter. The details for gl.polygonOffset() are shown here.

gl.polygonOffset(factor, units)
Specify the offset to be added to the z coordinate of each vertex drawn afterward.
The offset is calculated with the formula m * factor + r * units, where m represents the
inclination of the triangle with respect to the line of sight, and where r is the smallest
difference between two z coordinates values the hardware can distinguish.
Return value

None

Errors

None

Let’s look at the program Zfighting, which uses the polygon offset to reduce z fighting
(see Listing 7.11).
Listing 7.11

Zfighting.js

1 // Zfighting.js
...
23 function main() {
...
69
// Enable the polygon offset function
70
gl.enable(gl.POLYGON_OFFSET_FILL);
71
// Draw a rectangle
72
gl.drawArrays(gl.TRIANGLES, 0, n/2);
// The green triangle
73
gl.polygonOffset(1.0, 1.0);
// Set the polygon offset
74
gl.drawArrays(gl.TRIANGLES, n/2, n/2); // The yellow triangle
75 }
76
77 function initVertexBuffers(gl) {

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78
79
80
81
82
83
84
85
86
87
88

var verticesColors = new Float32Array([
// Vertex coordinates and color
0.0, 2.5, -5.0, 0.0, 1.0, 0.0, // The green triangle
-2.5, -2.5, -5.0, 0.0, 1.0, 0.0,
2.5, -2.5, -5.0, 1.0, 0.0, 0.0,
0.0, 3.0,
-3.0, -3.0,
3.0, -3.0,
]);
var n = 6;

-5.0,
-5.0,
-5.0,

1.0,
1.0,
1.0,

0.0,
1.0,
1.0,

0.0, // The yellow triangle
0.0,
0.0,

If you look at the program from line 80, you can see that the z coordinate for each vertex
is set to –5.0, so z fighting should occur.
Within the rest of the code, the polygon offset function is enabled at line 70. After
that, the green and yellow triangles are drawn at lines 72 and 74. For ease of reading,
the program uses only one buffer object, so gl.drawArrays() requires the second and
third arguments to be correctly set. The second argument represents the number of the
vertex to start from, while the third argument gives the number of vertices to be drawn.
Once the green triangle has been drawn, the polygon offset parameter is set using gl.
polygonOffset(). Subsequently, all the vertices drawn will have their z coordinate offset.
If you load this program, you will see the two triangles drawn correctly with no z fighting effects, as in Figure 7.28 (right side). If you now comment out line 73 and reload the
program, you will notice that z fighting occurs and looks similar to the left side of Figure
7.28.

Hello Cube
So far, the explanation of various WebGL features has been illustrated using simple triangles. You now have enough understanding of the basics to draw 3D objects. Let’s start by
drawing the cube shown in Figure 7.31. (The coordinates for each vertex are shown on the
right side.) The program used is called HelloCube, in which the eight vertices that define
the cube are specified using the following colors: white, magenta (bright reddish-violet),
red, yellow, green, cyan (bright blue), blue, and black. As was explained in Chapter 5,
“Using Colors and Texture Images,” because colors between the vertices are interpolated,
the resulting cube is shaded with an attractive color gradient (actually a “color solid,” an
analog of the two-dimensional “color wheel”).

Hello Cube

275

blue
v6(-1,1,-1)
v1(-1,1,1)
magenta

y
cyan
v5(1,1,-1)
v0(1,1,1) white

black v7(-1,-1,-1)

v2(-1,-1,1)
red

Figure 7.31

x
v4(1,-1,-1)
green

z

v3(1,-1,1) yellow

HelloCube and its vertex coordinates

Let’s consider the case where you would like to draw the cube like this with the command
you’ve been relying upon until now: gl.drawArrays(). In this case, you need to draw
using one of the following modes: gl.TRIANGLES, gl.TRIANGLE_STRIP, or gl.TRIANGLE_FAN.
The most simple and straightforward method would consist of drawing each face with
two triangles. In other words, you can draw a face defined by four vertices (v0, v1, v2, v3),
using two triangles defined by the two sets of three vertices (v0, v1, v2) and (v0, v2, v3),
respectively, and repeat the same process for all the other faces. In this case, the vertices
coordinates specified inside the buffer object would be these:
var vertices =
1.0, 1.0,
1.0, 1.0,
1.0, 1.0,
...
]);

new Float32Array([
1.0, -1.0, 1.0, 1.0, -1.0, -1.0, 1.0, // v0, v1, v2
1.0, -1.0, -1.0, 1.0, 1.0, -1.0, 1.0, // v0, v2, v3
1.0,
1.0, -1.0, 1.0, 1.0, -1.0, -1.0, // v0, v3, v4

Because one face is made up of two triangles, you need to know the coordinates of six
vertices to define it. There are six faces, so a total of 6×6 = 36 vertices are necessary. After
having specified the coordinates of each of the 36 vertices, write them in the buffer
object and then call gl.drawArrays(gl.TRIANGLES, 0, 36), which draws the cube. This
approach requires that you specify and handle 36 vertices, although the cube actually only
requires 8 unique vertices because several triangles share common vertices.
You could, however, take a more frugal approach by drawing a single face with gl.
TRIANGLE_FAN. Because gl.TRIANGLE_FAN allows you to draw a face defined by the 4-vertex
set (v0, v1, v2, v3), you end up only having to deal with a total of 4×6=24 vertices4.
However, you now need to call gl.drawArrays() separately for each face (six faces). So,
each of these two approaches has both advantages and drawbacks, but neither seems ideal.
4

You can cut down on the number of vertices using this kind of representation. It decreases the
number of necessary vertices to 14, which can be drawn with gl.TRIANGLE_STRIP.

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As you would expect, WebGL has a solution: gl.drawElements(). It’s an alternative way to
directly draw a three-dimensional object in WebGL, with a minimum of vertices. To use
this method, you will need the vertex coordinates of the entire object, which you will use
to explicitly describe how you want WebGL to draw the shape (the cube).
If we break our cube (see the right side of Figure 7.31) into vertices that constitute triangles, we get the structure shown in Figure 7.32. Looking at the left side of the figure,
you can see that Cube points to a Faces list, which, as the name implies, shows that the
cube is split into six faces: front, right, left, top, bottom, and back. In turn, each face is
composed of two triangles picked up from the Triangles list. The numbers in the Triangles
list represent the indices assigned to the Coordinate list. The vertex coordinates’ indices
are numbered in order starting from zero.

front

cube

right
up
left

0
1
2
0
2
3

up
down
v6(-1,1,-1)
v5(1,1,-1)
v1(-1,1,1)
v0(1,1,1)

r ig h t

v7(-1,-1,-1)

v4(1,-1,-1)

back

0
3
4
0
4
5

0

1, 1, 1

1, 1, 1

1

-1, 1, 1

1, 0, 1

2

-1, -1, 1

1, 0, 0

3

1, -1, 1

1, 1, 0

4

1, -1, -1

0, 1, 0

5

1, 1, -1

0, 1, 1

6

-1, 1, -1

0, 0, 1

7

-1, -1, -1

0, 0, 0

v2(-1,-1,1)
v3(1,-1,1)

fro n t

4
7
6
4
6
5

Figure 7.32 The associations of the faces that make up the cube, triangles, vertex
coordinates, and colors
This approach results in a data structure that describes the way the object (a cube) can be
built from its vertex and color data.

Drawing the Object with Indices and Vertices Coordinates
So far, you have been using gl.drawArrays() to draw vertices. However, WebGL supports
an alternative approach, gl.drawElements(), that looks similar to that of gl.draw
Arrays(). However, it has some advantages that we’ll explain later. First, let’s look at
how to use gl.drawElements(). You need to specify the indices based, not on gl.ARRAY_
BUFFER, but on gl.ELEMENT_ARRAY_BUFFER (introduced in the explanation of the buffer
object in Chapter 4). The key difference is that gl.ELEMENT_ARRAY_BUFFER handles data
structured by the indices.
Hello Cube

277

gl.drawElements(mode, count, type, offset)
Executes the shader and draws the geometric shape in the specified mode using the
indices specified in the buffer object bound to gl.ELEMENT_ARRAY_BUFFER.
Parameters

mode

Specifies the type of shape to be drawn (refer to Figure
3.17).
The following symbolic constants are accepted:
gl.POINTS, gl.LINE_STRIP, gl.LINE_LOOP, gl.LINES,
gl.TRIANGLE_STRIP, gl.TRIANGLE_FAN, or gl.TRIANGLES

count

Number of indices to be drawn (integer).

type

Specifies the index data type: gl.UNSIGNED_BYTE or gl.
UNSIGNED_SHORT5

offset

Specifies the offset in bytes in the index array where you
want to start rendering.

Return value

None

Errors

INVALID_ENUM

mode is none of the preceding values.

INVALID_VALUE

A negative value is specified for count or offset

Writing indices to the buffer object bound to gl.ELEMENT_ARRAY_BUFFER is done in the
same way you write the vertex information to the buffer object with gl.drawArrays().
That is to say, you use gl.bindBuffer() and gl.bufferData(), but the only difference is
that the first argument, target, is set to gl.ELEMENT_ARRAY_BUFFER. Let’s take a look at the
sample program.

Sample Program (HelloCube.js)
The sample program is shown in Listing 7.12. The vertex and fragment shaders set a quadrangular pyramid viewing volume and perform a perspective projection transformation
like ProjectiveView_mvpMatrix.js. It’s important to understand that gl.drawElements()
doesn’t do anything special. The vertex shader simply transforms the vertex coordinates,
and the fragment shader sets the color passed by the varying variable to gl_FragColor.
The key difference from the previous programs comes down to the processing of the
buffer object in initVertexBuffers().

5

Even if type doesn’t correspond to the type (Uint8Array or Uint16Array) of the data array specified
in gl.ELEMENT_ARRAY_BUFFER, no error is returned. However, if, for example, you specify the index
with a Uint16Array type, and set type to gl.UNSIGNED_BYTE, in some cases, the object might not
be completely displayed.

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Listing 7.12

HelloCube.js

1 // HelloCube.js
2 // Vertex shader program
3 var VSHADER_SOURCE =
...
8
'void main() {\n' +
9
' gl_Position = u_MvpMatrix * a_Position;\n' +
10
' v_Color = a_Color;\n' +
11
'}\n';
12
13 // Fragment shader program
14 var FSHADER_SOURCE =
...
19
'void main() {\n' +
20
' gl_FragColor = v_Color;\n' +
21
'}\n';
22
23 function main() {
...
40
// Set the vertex coordinates and color
41
var n = initVertexBuffers(gl);
...
47
// Set the clear color and enable the hidden surface removal
48
gl.clearColor(0.0, 0.0, 0.0, 1.0);
49
gl.enable(gl.DEPTH_TEST);
...
58
// Set the eye point and the viewing volume
59
var mvpMatrix = new Matrix4();
60
mvpMatrix.setPerspective(30, 1, 1, 100);
61
mvpMatrix.lookAt(3, 3, 7, 0, 0, 0, 0, 1, 0);
62
63
// Pass the model view projection matrix to u_MvpMatrix
64
gl.uniformMatrix4fv(u_MvpMatrix, false, mvpMatrix.elements);
65
66
// Clear the color and depth buffer
67
gl.clear(gl.COLOR_BUFFER_BIT | gl.DEPTH_BUFFER_BIT);
68
69
// Draw the cube
70
gl.drawElements(gl.TRIANGLES, n, gl.UNSIGNED_BYTE, 0);
71 }
72
73 function initVertexBuffers(gl) {
...
82
var verticesColors = new Float32Array([

Hello Cube

279

83
84
85
86

// Vertex coordinates
1.0, 1.0, 1.0,
-1.0, 1.0, 1.0,
-1.0, -1.0, 1.0,

and color
1.0, 1.0,
1.0, 0.0,
1.0, 0.0,

1.0,
1.0,
0.0,

// v0 White
// v1 Magenta
// v2 Red

0.0,

0.0

// v7 Black

...
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106

-1.0, -1.0, -1.0,
]);

0.0,

// Indices of the vertices
var indices = new Uint8Array([
0, 1, 2,
0, 2, 3,
// front
0, 3, 4,
0, 4, 5,
// right
0, 5, 6,
0, 6, 1,
// up
1, 6, 7,
1, 7, 2,
// left
7, 4, 3,
7, 3, 2,
// down
4, 7, 6,
4, 6, 5
// back
]);

// Create a buffer object
var vertexColorBuffer = gl.createBuffer();
var indexBuffer = gl.createBuffer();
...
111
// Write the vertex coordinates and color to the buffer object
112
gl.bindBuffer(gl.ARRAY_BUFFER, vertexColorBuffer);
113
gl.bufferData(gl.ARRAY_BUFFER, verticesColors, gl.STATIC_DRAW);
114
115
var FSIZE = verticesColors.BYTES_PER_ELEMENT;
116
// Assign the buffer object to a_Position and enable it
117
var a_Position = gl.getAttribLocation(gl.program, 'a_Position');
...
122
gl.vertexAttribPointer(a_Position, 3, gl.FLOAT, false, FSIZE * 6, 0);
123
gl.enableVertexAttribArray(a_Position);
124
// Assign the buffer object to a_Position and enable it
125
var a_Color = gl.getAttribLocation(gl.program, 'a_Color');
...
130
gl.vertexAttribPointer(a_Color, 3, gl.FLOAT, false, FSIZE * 6, FSIZE * 3);
131
gl.enableVertexAttribArray(a_Color);
132
133
// Write the indices to the buffer object
134
gl.bindBuffer(gl.ELEMENT_ARRAY_BUFFER, indexBuffer);
135
gl.bufferData(gl.ELEMENT_ARRAY_BUFFER, indices, gl.STATIC_DRAW);
136
137
return indices.length;
138 }

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The processing flow in the JavaScript main() is the same as in ProjectiveView_mvpMatrix.
js, but let’s quickly review it. After having written the vertex data in the buffer object
through a call to initVertexBuffers() at line 41, you enable the hidden surface removal
function at line 49. This is necessary to allow WebGL to correctly draw the cube, taking
into consideration the relationship between the front and the back faces.
You set the eye point and the viewing volume from line 59 to line 61 and pass the model
view projection matrix to the vertex shader’s uniform variable u_MvpMatrix.
At line 67, you clear the color and depth buffers and then draw the cube using
gl.drawElements() at line 70. The use of gl.drawElements() in this program is the

main difference to ProjectiveView_mvpMatrix.js, so let’s take a look at that.

Writing Vertex Coordinates, Colors, and Indices to the Buffer Object
The method to assign the vertex coordinates and the color information to the attribute
variable using the buffer object in initVertexBuffers() is unchanged. This time, because
you won’t necessarily use the vertex information in the order specified in the object
buffer, you need to additionally specify in which order you will use it. For that you will
use the vertex order specified in verticesColors as indices. In short, the vertex information specified first in the buffer object will be set to index 0, the vertex information specified in second place in the buffer object will be set to index 1, and so on. Here, we show
the part of the program that specifies the indices in initVertexBuffers():
73 function initVertexBuffers(gl) {
...
82
var verticesColors = new Float32Array([
83
// Vertex coordinates and color
84
1.0, 1.0, 1.0,
1.0, 1.0, 1.0, // v0 White
85
-1.0, 1.0, 1.0,
1.0, 0.0, 1.0, // v1 Magenta
...
91
-1.0, -1.0, -1.0,
0.0, 0.0, 0.0
// v7 Black
92
]);
93
94
// Indices of the vertex coordinates
95
var indices = new Uint8Array([
96
0, 1, 2,
0, 2, 3,
// front
97
0, 3, 4,
0, 4, 5,
// right
98
0, 5, 6,
0, 6, 1,
// up
99
1, 6, 7,
1, 7, 2,
// left
100
7, 4, 3,
7, 3, 2,
// down
101
4, 7, 6,
4, 6, 5
// back
102 ]);
103
104
// Create a buffer object
105
var vertexColorBuffer = gl.createBuffer();

Hello Cube

281

106

var indexBuffer = gl.createBuffer();
...
136
// Write the indices to the buffer object
137
gl.bindBuffer(gl.ELEMENT_ARRAY_BUFFER, indexBuffer);
138
gl.bufferData(gl.ELEMENT_ARRAY_BUFFER, indices, gl.STATIC_DRAW);
139
140
return indices.length;
141 }

As you may have noticed, at line 106, you create the buffer object (indexBuffer) in which
to write the indices. These indices are stored in the array indices at line 95. Because the
indices are integers (0, 1, 2, ...), you use an integer typed array Uint8Array (unsigned
8-bit encoded integer). If there are more than 256 indices, use Uint16Array instead. The
content of this array is the triangles list of Figure 7.33, where each grouping of three
indices points to the three vertex coordinates for that triangle. Generally, this index
doesn’t need to be manually created because 3D modeling tools, introduced in the next
chapter, usually generate it along with the vertices information.

gl.ELEMENT_ARRAY_BUFFER

cube

front
right
up
left
down

v6(-1,1,-1)
v5(1,1,-1)

back

v1(-1,1,1)
v0(1,1,1)
v7(-1,-1,-1)
v4(1,-1,-1)

0
1
2
0
2
3
0
3
4
0
4
5

gl.ARRAY_BUFFER

0

1, 1, 1

1, 1, 1

1

-1, 1, 1

1, 0, 1

2

-1, -1, 1

1, 0, 0

3

1, -1, 1

1, 1, 0

4

1, -1, -1

0, 1, 0

5

1, 1, -1

0, 1, 1

6

-1, 1, -1

0, 0, 1

7

-1, -1, -1

0, 0, 0

v2(-1,-1,1)
v3(1,-1,1)

Figure 7.33

Contents of gl.ELEMENT_ARRAY_BUFFER and gl.ARRAY_BUFFER

The setup for the specified indices is performed at lines 134 and 135. This is similar to the
way buffer objects have been written previously, with the difference that the first argument is modified to gl.ELEMENT_ARRAY_BUFFER. This is to let the WebGL system know that
the contents of the buffer are indices.
Once executed, the internal state of the WebGL system is as detailed in Figure 7.34.

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a ttr ib u te v a r ia b le

JavaScript

gl . ELEMENT_ARRAY_BUFFER

fu n c tio n m a in ( ) {
v a r g l= g e tW e b G L …
…
g l.b in d B u ffe r ( … ) ;

Figure 7.34

B u f f e r O b je c t
0,
0,
0,
1,
7,
4,

1,
3,
5,
6,
4,
7,

2,
4,
6,
7,
3,
6,

0,
0,
0,
1,
7,
4,

2,
4,
6,
7,
3,
6,

3,
5,
1,
2,
2,
5

gl .ARRAY_BUFFER

B u f f e r O b je c t
1, 1, 1,
-1 , 1 , 1 ,
-1 , -1 , 1 ,
1 , -1 , 1 ,
1 , -1 , -1 ,
1 , 1 , -1 ,
-1 , 1 , -1 ,
-1 , -1 , -1 ,

1,
1,
1,
1,
0,
0,
0,
0,

1,
0,
0,
1,
1,
1,
0,
0,

F ra g m e n t
Shader

1,
1,
0,
0,
0,
1,
1,
0

gl.ELEMENT_ARRAY_BUFFER and gl.ARRAY_BUFFER

Once set up, the call to gl.drawElements() at line 70 draws the cube:
69
70

// Draw the cube
gl.drawElements(gl.TRIANGLES, n, gl.UNSIGNED_BYTE, 0);

You should note that the second argument of gl.drawElements(), the number of indices,
represents the number of vertex coordinates involved in the drawing, but it is not identical to the number of vertices coordinates written to gl.ARRAY_BUFFER.
When you call gl.drawElements(), the indices are extracted from the buffer object
(indexBuffer) bound to gl.ELEMENT_ARRAY_BUFFER, while the associated vertex information is retrieved from the buffer object (vertexColorBuffer) bound to gl.ARRAY_BUFFER.
All these pieces of information are then passed to the attribute variable. The process is
repeated for each index, and then the whole cube gets drawn by a single call to
gl.drawElements(). With this approach, because you refer to the vertex information
through indices, you can recycle the vertex information. Although gl.drawElements()
allows you to curb memory usage by sharing the vertex information, the cost is a process
to convert the indices to vertex information (that is, a level of indirection). This means
that the choice between gl.drawElements() and gl.drawArrays(), because they both
have pros and cons, will eventually depend on the system implementation.
At this stage, although it’s clear that gl.drawElements() is an efficient way to draw 3D
shapes, one key feature is missing. There is no way to control color, so it is helpful to draw
a cube using a single solid color, as shown in Figure 7.31.
For example, let’s consider the case where you would like to modify the color of each face
of the cube, as shown in Figure 7.35, or map textures to the faces. You need to know the
color or texture information for each face, yet you cannot implement this with the combination of indices, triangle list, and vertex coordinates shown in Figure 7.33.

Hello Cube

283

Figure 7.35

Cube with differently colored faces

In the following section, we will examine how to address this problem and specify the
color information for each face.

Adding Color to Each Face of a Cube
As discussed before, you can only pass per-vertex information to the vertex shader. This
implies that you need to pass the face’s color and the vertices of the triangles as vertex
information to the vertex shader. For instance, to draw the “front” face in blue, made up
of v0, v1, v2, and v3 (Figure 7.33), you need to specify the same blue color for each of the
vertices.
However, as you may have noticed, v0 is also shared by the “right” and “top” faces as well
as the “front” face. Therefore, if you specify the color blue for the vertices that form the
“front” face, you are then unable to choose a different color for those vertices that also
belong to another face. To cope with this problem, although this might not seem as efficient, you must create duplicate entries for the shared vertices in the vertices coordinates
listing, as illustrated in Figure 7.36. Doing so, you will have to handle common vertices
with identical coordinates in the face’s triangle list as separate entities.6

6

If you break down all the faces into triangles and draw using gl.drawArrays(), you have
to process 6 vertices * 6 faces = 36 vertices, so the difference between gl.drawArrays() and
gl.drawElements() in memory usage is negligible. This is because a cube or a cuboid is a special
3D object whose faces are connected vertically; therefore, each vertex needs to have three colors.
However, in the case of complex 3D models, specifying several colors to a single vertex would be rare.

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front

cube

right
up
left
down
back

u p (re d )
v6(-1,1,-1)

v5(1,1,-1)
v1(-1,1,1)

0
1
2
0
2
3
4
5

0

x0, y0, z0

0, 0, 1

1

x1, y1, z1

0, 0, 1

2

x2, y2, z2

0, 0, 1

3

x3, y3, z3

0, 0, 1

4

x0, y0, z0

0, 1, 0

5

x3, y3, z3

0, 1, 0

6

x4, y4, z4

0, 1, 0

7

x5, y5, z5

0, 1, 0

…

…

20

x4, y4, z4

0, 1, 1

21

x7, y7, z7

0, 1, 1

22

x6, y6, z6

0, 1, 1

23

x5, y5, z5

0, 1, 1

6
4
6
7

v0(1,1,1)
v7(-1,-1,-1)

r ig h t ( g r e e n )
v4(1,-1,-1)

v2(-1,-1,1)
v3(1,-1,1)

20
21
22
20
22
23

f r o n t ( b lu e )

Figure 7.36 The faces that constitute the cube, the triangles, and the relationship between
vertices coordinates (configured so that you can choose a different color for each face)
When opting for such a configuration, the contents of the index list, which consists of
the face’s triangle list, will differ from face to face, thus allowing you to modify the color
for each face. This approach can also be used if you want to map a texture to each face.
You would need to specify the texture coordinates for each vertex, but you can actually
deal with this by rewriting the color list (Figure 7.36) as texture coordinates. The sample
program in the section “Rotate Object” in Chapter 10 covers this approach in more detail.
Let’s take a look at the sample program ColoredCube, which displays a cube with each face
painted a different color. The screen shot of ColoredCube is identical to Figure 7.35.

Sample Program (ColoredCube.js)
The sample program is shown in Listing 7.13. Because the only difference from
HelloCube.js is the method of storing vertex information into the buffer object, let’s look
in more detail at the code related to the initVertexBuffers(). The main differences to
HelloCube.js are

• In HelloCube.js, the vertex coordinates and color are stored in a single buffer
object, but because this make the array unwieldy, the program has been modified so
that they are now stored in separate buffer objects.

• The respective contents of the vertex array (which stores the vertex coordinates), the
color array (which stores the color information), and the index array (which stores
the indices) are modified in accordance with the configuration described in Figure
7.36 (lines 83, 92, and 101).

Hello Cube

285

• To keep the sample program as compact as possible, the function
initArrayBuffer() is defined, which bundles the buffer object creation, binding,
writing of data, and enabling (lines 116, 119, and 129).

As you examine the program, take note of how the second bullet is implemented to match
the structure shown in Figure 7.36.
Listing 7.13

ColoredCube.js

1 // ColoredCube.js
...
23 function main() {
...
40
// Set the vertex information
41
var n = initVertexBuffers(gl);
...
69
// Draw the cube
70
gl.drawElements(gl.TRIANGLES, n, gl.UNSIGNED_BYTE, 0);
71 }
72
73 function initVertexBuffers(gl) {
...
83
var vertices = new Float32Array([
// Vertex coordinates
84
1.0, 1.0, 1.0, -1.0, 1.0, 1.0, -1.0,-1.0, 1.0,
1.0,-1.0, 1.0,
85
1.0, 1.0, 1.0,
1.0,-1.0, 1.0,
1.0,-1.0,-1.0,
1.0, 1.0,-1.0,
86
1.0, 1.0, 1.0,
1.0, 1.0,-1.0, -1.0, 1.0,-1.0, -1.0, 1.0, 1.0,
...
89
1.0,-1.0,-1.0, -1.0,-1.0,-1.0, -1.0, 1.0,-1.0,
1.0, 1.0,-1.0
90
]);
91
92
var colors = new Float32Array([
// Colors
93
0.4, 0.4, 1.0, 0.4, 0.4, 1.0, 0.4, 0.4, 1.0, 0.4, 0.4, 1.0,
94
0.4, 1.0, 0.4, 0.4, 1.0, 0.4, 0.4, 1.0, 0.4, 0.4, 1.0, 0.4,
95
1.0, 0.4, 0.4, 1.0, 0.4, 0.4, 1.0, 0.4, 0.4, 1.0, 0.4, 0.4,
...
98
0.4, 1.0, 1.0, 0.4, 1.0, 1.0, 0.4, 1.0, 1.0, 0.4, 1.0, 1.0
99
]);
100
101
var indices = new Uint8Array([
// Indices of the vertices
102
0, 1, 2,
0, 2, 3,
// front
103
4, 5, 6,
4, 6, 7,
// right
104
8, 9,10,
8,10,11,
// up
...
107
20,21,22, 20,22,23
// back
108

286

]);

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Toward the 3D World

109
110
111

// Create a buffer object
var indexBuffer = gl.createBuffer();
...
115
// Write the vertex coordinates and color to the buffer object
116
if (!initArrayBuffer(gl, vertices, 3, gl.FLOAT, 'a_Position'))
117
return -1;
118
119
if (!initArrayBuffer(gl, colors, 3, gl.FLOAT, 'a_Color'))
120
return -1;
...
122
// Write the indices to the buffer object
123
gl.bindBuffer(gl.ELEMENT_ARRAY_BUFFER, indexBuffer);
124
gl.bufferData(gl.ELEMENT_ARRAY_BUFFER, indices, gl.STATIC_DRAW);
125
126
return indices.length;
127 }
128
129 function initArrayBuffer(gl, data, num, type, attribute) {
130
var buffer = gl.createBuffer();
// Create a buffer object
...
135
// Write date into the buffer object
136
gl.bindBuffer(gl.ARRAY_BUFFER, buffer);
137
gl.bufferData(gl.ARRAY_BUFFER, data, gl.STATIC_DRAW);
138
// Assign the buffer object to the attribute variable
139
var a_attribute = gl.getAttribLocation(gl.program, attribute);
...
144
gl.vertexAttribPointer(a_attribute, num, type, false, 0, 0);
145
// Enable the assignment of the buffer object to the attribute variable
146
gl.enableVertexAttribArray(a_attribute);
147
148
return true;
149 }

Experimenting with the Sample Program
In ColoredCube, you specify a different color for each face. So what happens when you
choose an identical color for all the faces? For example, let’s try to set the color information in ColoredCube.js’s colors array to “white,” as shown next. We will call this
program ColoredCube_singleColor.js:
1
92
93

// ColoredCube_singleColor.js
...
var colors = new Float32Array([
1.0, 1.0, 1.0, 1.0, 1.0, 1.0,

1.0, 1.0, 1.0,

1.0, 1.0, 1.0,

Hello Cube

287

94
98
99

1.0, 1.0, 1.0,
...
1.0, 1.0, 1.0,
]);

1.0, 1.0, 1.0,

1.0, 1.0, 1.0,

1.0, 1.0, 1.0,

1.0, 1.0, 1.0,

1.0, 1.0, 1.0,

1.0, 1.0, 1

When you execute the program, you see an output like the screenshot shown in Figure
7.37. One result of using a single color is that it becomes difficult to actually recognize
the cube. Up until now you could differentiate each face because they were differently
colored; therefore, you could recognize the whole shape as a solid. However, when you
switch to a unique color, you lose this three-dimensional impression.

Figure 7.37

Cube with its faces being identically colored

In contrast, in the real world, when you put a white box on a table, you can identify it as
a solid (see Figure 7.38). This is because each face, although the same white color, presents
a slightly different appearance because each is lit slightly differently. In ColoredCube_
singleColor, such an effect is not programmed, so the cube is hard to recognize. We will
explore how to correctly light 3D scenes in the next chapter.

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Toward the 3D World

Figure 7.38

White box in the real world

Summary
In this chapter, through the introduction of the depth information, you have examined
setting the viewer’s eye point and viewing volume, looked at how to draw real 3D objects,
and briefly examined the local and world coordinate system. Many of the examples were
similar to those previously explained for the two-dimensional world, except for the introduction of the z-axis to handle depth information.
The next chapter explains how to light 3D scenes and how to draw and manipulate
three-dimensional shapes with complex structures. We will also return to the function
initShaders(), which has hidden a number of complex issues that you now have enough
understanding to explore.

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Chapter 8

Lighting Objects

This chapter focuses on lighting objects, looking at different light sources and
their effects on the 3D scene. Lighting is essential if you want to create realistic
3D scenes because it helps to give the scene a sense of depth.
The following key points are discussed in this chapter:

• Shading, shadows, and different types of light sources including point,
directional, and ambient

• Reflection of light in the 3D scene and the two main types: diffuse and
ambient

• The details of shading and how to implement the effect of light to make
objects, such as the pure white cube in the previous chapter, look threedimensional
By the end of this chapter, you will have all the knowledge you need to create
lighted 3D scenes populated with both simple and complex 3D objects.

Lighting 3D Objects
When light hits an object in the real world, part of the light is reflected by the
surface of the object. Only after this reflected light enters your eyes can you see
the object and distinguish its color. For example, a white box reflects white light
which, when it enters your eyes, allows you to tell that the box is white.

In the real world, two important phenomena occur when light hits an object
(see Figure 8.1):

• Depending on the light source and direction, surface color is shaded.
• Depending on the light source and direction, objects “cast” shadows on the ground
or the floor.

Shadowing

Shading
Figure 8.1

Shading and shadowing

In the real world, you usually notice shadows, but you quite often don’t notice shading,
which gives 3D objects their feeling of depth. Shading is subtle but always present. As
shown in Figure 8.1, even surfaces of a pure white cube are distinguishable because each
surface is shaded differently by light. As you can see, the surfaces hit by more light are
brighter, and the surfaces hit by less light are darker, or more shaded. These differences
allow you to distinguish each surface and ensure that the cube looks cubic.
In 3D graphics, the term shading1 is used to describe the process that re-creates this
phenomenon where the colors differ from surface to surface due to light. The other
phenomenon, that the shadow of an object falls on the floor or ground, is re-created using
a process called shadowing. This section discusses shading. Shadowing is discussed in
Chapter 10, which focuses on a set of useful techniques that build on your basic knowledge of WebGL.

1

Shading is so critical to 3D graphics that the core language, GLSL ES, is a shader language, the
OpenGL ES Shading Language. The original purpose of shaders was to re-create the phenomena of
shading.

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When discussing shading, you need to consider two things:

• The type of light source that is emitting light
• How the light is reflected from surfaces of an object and enters the eye
Before we begin to program, let’s look at different types of light sources and how light is
reflected from different surfaces.

Types of Light Source
When light illuminates an object, a light source emits the light. In the real world, light
sources are divided into two main categories: directional light, which is something like
the sun that emits light naturally, and point light, which is something like a light bulb
that emits light artificially. In addition, there is ambient light that represents indirect
light (that is, light emitted from all light sources and reflected by walls or other objects
(see Figure 8.2). In 3D graphics, there are additional types of light sources. For example,
there is a spot light representing flashlights, headlamps, and so on. However, in this book,
we don’t address these more specialized light sources. Refer to the book OpenGL ES 2.0
Programming Guide for further information on these specialized light sources.

Directional light

Figure 8.2

Point light

Ambient light

Directional light, point light, and ambient light

Focusing on the three main types of light source covered in this book:
Directional light: A directional light represents a light source whose light rays are parallel. It is a model of light whose source is considered to be at an infinite distance, such
as the sun. Because of the distance travelled, the rays are effectively parallel by the time
they reach the earth. This light source is considered the simplest, and because its rays are
parallel can be specified using only direction and color.
Point light: A point light represents a light source that emits light in all directions from
one single point. It is a model of light that can be used to represent light bulbs, lamps,

Lighting 3D Objects

293

flames, and so on. This light source is specified by its position and color.2 However, the
light direction is determined from the position of the light source and the position at
which the light strikes a surface. As such, its direction can change considerably within
the scene.
Ambient light: Ambient light (indirect light) is a model of light that is emitted from
the other light source (directional or point), reflected by other objects such as walls, and
reaches objects indirectly. It represents light that illuminates an object from all directions
and has the same intensity.3 For example, if you open the refrigerator door at night, the
entire kitchen becomes slightly lighter. This is the effect of the ambient light. Ambient
light does not have position and direction and is specified only by its color.
Now that you know the types of light sources that illuminate objects, let’s discuss how
light is reflected by the surface of an object and then enters the eye.

Types of Reflected Light
How light is reflected by the surface of an object and thus what color the surface will
become is determined by two things: the type of the light and the type of surface of the
object. Information about the type of light includes its color and direction. Information
about the surface includes its color and orientation.
When calculating reflection from a surface, there are two main types: diffuse reflection
and environment (or ambient) reflection. The remainder of this section describes how to
calculate the color due to reflection using the two pieces of information described earlier.
There is a little bit of math to be considered, but it’s not complicated.
Diffuse Reflection
Diffuse reflection is the reflection of light from a directional light or a point light. In
diffuse reflection, the light is reflected (scattered) equally in all directions from where
it hits (see Figure 8.3). If a surface is perfectly smooth like a mirror, all incoming light
is reflected; however, most surfaces are rough like paper, rock, or plastic. In such cases,
the light is scattered in random directions from the rough surface. Diffuse reflection is a
model of this phenomenon.

2

This type of light actually attenuates; that is, it is strong near the source and becomes weaker farther
from the source. For the sake of simplicity of the description and sample programs, light is treated
as nonattenuating in this book. For attenuation, please refer to the book OpenGL ES 2.0 Programming
Guide.

3

In fact, ambient light is the combination of light emitted from light sources and reflected by various
surfaces. It is approximated in this way because it would otherwise need complicated calculations to
take into account all the many light sources and how and where they are reflected.

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orientation of
the surface

θ

light
source

orientation of
the surface

light
source

light
direction

A
diffuse reflection

Figure 8.3

B
C
diffuse reflection (reflection differs by light direction)

Diffuse reflection

In diffuse reflection, the color of the surface is determined by the color and the direction
of light and the base color and orientation of the surface. The angle between the light
direction and the orientation of the surface is defined by the angle formed by the light
direction and the direction “perpendicular” to the surface. Calling this angle θ, the surface
color by diffuse reflection is calculated using the following formula.
Equation 8.1
〈 surface color by diffuse reflection 〉 =
〈light color 〉 × 〈base color of surface〉 × cosθ
where  is the color of light emitted from a directional light or a point light.
Multiplication with the  is performed for each RGB component
of the color. Because light by diffuse reflection is scattered equally in all directions from
where it hits, the intensity of the reflected light at a certain position is the same from
any angle (see Figure 8.4).

orientation of the surface

Figure 8.4

The intensity of light at a given position is the same from any angle

Ambient Reflection
Ambient reflection is the reflection of light from another light source. In ambient reflection, the light is reflected at the same angle as its incoming angle. Because an ambient
light illuminates an object equally from all directions with the same intensity, its brightness is the same at any position (see Figure 8.5). It can be approximated as follows.
Lighting 3D Objects

295

Equation 8.2
〈 surface color by ambient reflection 〉 =
〈light color 〉 × 〈base color of surface〉
where  is the color of light emitted from other light source.
orientation of the surface

ambient reflection

Figure 8.5

ambient reflection (the same reflection at any position)

Ambient reflection

When both diffuse reflection and ambient reflection are present, the color of the surface is
calculated by adding, as follows.
Equation 8.3
〈 surface color by diffuse and ambient reflection 〉 =
〈 surface color by diffuse reflection 〉 + 〈 surface color by ambient reflection 〉
Note that it is not required to always use both light sources, or use the formulas exactly
as mentioned here. You are free to modify each formula to achieve the effect you require
when showing the object.
Now let’s construct some sample programs that perform shading (shading and coloring
the surfaces of an object by placing a light source at an appropriate position). First let’s try
to implement shading due to directional light and its diffuse reflection.

Shading Due to Directional Light and Its Diffuse Reflection
As described in the previous section, surface color is determined by light direction and the
orientation of the surface it strikes when considering diffuse reflection. The calculation of
the color due to directional light is easy because its direction is constant. The formula for
calculating the color of a surface by diffuse reflection (Equation 8.1) is shown again here:
〈 surface color by diffuse reflection 〉 =
〈light color 〉 × 〈base color of surface〉 × cosθ
The following three pieces of information are used:

• The color of the light source (directional light)
• The base color of the surface
• The angle (θ) between the light and the surface
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The color of a light source may be white, such as sunlight, or other colors, such as the
orange of lighting in road tunnels. As you know, it can be represented by RGB. White
light such as sunlight has an RGB value of (1.0, 1.0, 1.0). The base color of a surface
means the color that the surface was originally defined to have, such as red or blue. To
calculate the color of a surface, you need to apply the formula for each of the three RGB
components; the calculation is performed three times.
For example, assume that the light emitted from a light source is white (1.0, 1.0, 1.0), and
the base color of the surface is red (1.0, 0.0, 0.0). From Equation 8.1, when θ is 0.0 (that is,
when the light hits perpendicularly), cos θ becomes 1.0. Because the R component of the
light source is 1.0, the R component of the base surface color is 1.0, and the cos θ is 1.0,
the R component of the surface color by diffuse reflection is calculated as follows:
R = 1.0 * 1.0 * 1.0 = 1.0
The G and B components are also calculated in the same way, as follows:
G = 1.0 * 0.0 * 1.0 = 0.0
B = 1.0 * 0.0 * 1.0 = 0.0
From these calculations, when white light hits perpendicularly on a red surface, the
surface color by diffuse reflection turns out to be (1.0, 0.0, 0.0), or red. This is consistent
with real-world experience. Conversely, when the color of the light source is red and the
base color of a surface is white, the result is the same.
Let’s now consider the case when θ is 90 degrees, or when the light does not hit the
surface at all. From your real-world experience, you know that in this case the surface will
appear black. Let’s validate this. Because cos θ is 0 when θ is 90 degrees, and anything
multiplied by zero is zero, the result of the formula is 0 for R, G, and B; that is, the surface
color becomes (0.0, 0.0, 0.0), or black, as expected. Equally, when θ is 60 degrees, you’d
expect that a small amount of light falling on a red surface would result in a darker red
color, and because cos θ is 0.5, the surface color is (0.5, 0.0, 0.0), which is dark red, as
expected.
These simple examples have given you a good idea of how to calculate surface color due
to diffuse reflection. To allow you to factor in directional light, let’s transform the preceding formula to make it easy to handle so you can then explore how to draw a cube lit by
directional light.

Calculating Diffuse Reflection Using the Light Direction and the
Orientation of a Surface
In the previous examples, an arbitrary value for θ was chosen. However, typically it is
complicated to get the angle θ between the light direction and the orientation of a surface.
For example, when creating a model, the angle at which light hits each surface cannot
be determined in advance. In contrast, the orientation of each surface can be determined

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297

regardless of where light hits from. Because the light direction is also determined when
its light source is determined, it seems convenient to try to use these two pieces of
information.
Fortunately, mathematics tells us that cos θ is derived by calculating the dot product of
the light direction and the orientation of a surface. Because the dot product is so often
used, GLSL ES provides a function to calculate it.4 (More details can be found in Appendix
B, “Built-In Functions of GLSL ES 1.0.”) When representing the dot product by “·”, cos θ is
defined as follows:
cosθ = 〈light direction 〉 i 〈orientation of a surface〉
From this, Equation 8.1 can be transformed as following Equation 8.4:
Equation 8.4
〈 surface color by diffuse reflection 〉 =
〈light color 〉 × 〈base color of surface〉 ×

( 〈light

direction 〉 i 〈orientation of a surface〉 )

Here, there are two points to be considered: the length of the vector and the light direction. First, the length of vectors that represent light direction and orientation of the
surface, such as (2.0, 2.0, 1.0), must be 1.0,5 or the color of the surface may become too
dark or bright. Adjusting the components of a vector so that its length becomes 1.0 is
called normalization.6 GLSL ES provides functions for normalizing vectors that you can
use directly.
The second point to consider concerns the light direction for the reflected light. The light
direction is the opposite direction from that which the light rays travel (see Figure 8.6).

4

Mathematically, the dot product of two vectors n and l is written as follows:
n • 1 = |n| x |1| x cos θ
where || means the length of the vector. From this equation, you can see that when the lengths of n
and l are 1.0, the dot product is equal to cos θ. If n is (nx, ny, nz) and l is (lx, ly, lz), then nl = nx * lx + ny*
ly+ nz * lz from the law of cosines.

5

If the components of the vector n are (nx, ny, nz), its length is as follows:

length of n = | n | =
6

nx 2 + ny 2 + nz 2

Normalized n is (nx/m, ny/m, nz/m), where m is the length of n. |n| = sqrt(9) = 3. The vector (2.0, 2.0,
1.0) above is normalized into (2.0/3.0, 2.0/3.0, 1.0/3.0).

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orientation of
the surface

θ

light
source
light
direction

diffuse reflection

Figure 8.6

The light direction is from the reflecting surface to the light source

Because we aren’t using an angle to specify the orientation of the surface, we need another
mechanism to do that. The solution is to use normal vectors.

The Orientation of a Surface: What Is the Normal?
The orientation of a surface is specified by the direction perpendicular to the surface and
is called a normal or a normal vector. This direction is represented by a triple number,
which is the direction of a line from the origin (0, 0, 0) to (nx, ny, nz) specified as the
normal. For example, the direction of the normal (1, 0, 0) is the positive direction of the
x-axis, and the direction of the normal (0, 0, 1) is the positive direction of the z-axis.
When considering surfaces and their normals, two properties are important for our
discussion.
A Surface Has Two Normals
Because a surface has a front face and a back face, each side has its own normal; that is,
the surface has two normals. For example, the surface perpendicular to the z-axis has a
front face that is facing toward the positive direction of the z-axis and a back face that is
facing the negative direction of the z-axis, as shown in Figure 8.7. Their normals are (0, 0,
1) and (0, 0, –1), respectively.
y

y
v1

v0

x

x

v2

z

v3
th e n o rm a l (0 , 0 , 1 )

Figure 8.7

z
the normal (0, 0, 1)

Normals
Lighting 3D Objects

299

In 3D graphics, these two faces are distinguished by the order in which the vertices are
specified when drawing the surface. When you draw a surface specifying vertices in the
order7 v0, v1, v2, and v3, the front face is the one whose vertices are arranged in a clockwise fashion when you look along the direction of the normal of the face (same as the
right-handed rule determining the positive direction of rotation in Chapter 3, “Drawing
and Transforming Triangles”). So in Figure 8.7, the front face has the normal (0, 0, –1) as
in the right side of the figure.
The Same Orientation Has the Same Normal
Because a normal just represents direction, surfaces with the same orientation have the
same normal regardless of the position of the surfaces.
If there is more than one surface with the same orientation placed at different positions,
the normals of these surfaces are identical. For example, the normals of a surface perpendicular to the z-axis, whose center is placed at (10, 98, 9), are still (0, 0, 1) and (0, 0, –1).
They are the same as when it is positioned at the origin (see Figure 8.8).
y

y

v1

v0

x

x

v2

z

v3

z

th e n o rm a l (0 , 0 , 1 )

th e n o rm a l (0 , 0 , -1 )

Figure 8.8 If the orientation of the surface is the same, the normal is identical regardless of
its position
The left side of Figure 8.9 shows the normals that are used in the sample programs in this
section. Normals are labeled using, for example “n(0, 1, 0)” as in this figure.

7

Actually, this surface is composed of two triangles: a triangle drawn in the order v0, v1, and v2, and a
triangle drawn in the order v0, v2, and v3.

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y
v6(-1, 1,-1)

y
n(0, 1, 0)

n(0, 1, 0)

v6(-1, 1,-1)
v1(-1,1,1)

v5(1, 1,-1)
v1(-1,1,1)

n(0, 0, -1)

v0(1, 1, 1)

v0(1, 1, 1)

n(-1, 0, 0) v7(-1,-1,-1)
n(0, 0, 1)

z

v7(-1,-1,-1)

n(1, 0, 0)
v4(1,-1,-1)

v2(-1,-1,1)

v5(1, 1,-1)

x

n(1, 0, 0)

n(0, 0, 1)
v4(1,-1,-1)

x

v2(-1,-1,1)

v3(1,-1,1)

z

v3(1,-1,1)

n(0, -1, 0)
(all normals are not displayed.)

Figure 8.9

Normals of the surfaces of a cube

Once you have calculated the normals for a surface, the next task is to pass that data to
the shader programs. In the previous chapter, you passed color data for a surface to the
shader as “per-vertex data.” You can pass normal data using the same approach: as pervertex data stored in a buffer object. In this section, as shown in Figure 8.9 (right side),
the normal data is specified for each vertex, and in this case there are three normals per
vertex, just as there are three color data specified per vertex.8
Now let’s construct a sample program LightedCube that displays a red cube lit by a white
directional light. The result is shown in Figure 8.10.

Figure 8.10
8

LightedCube

Cubes or cuboids are simple but special objects whose three surfaces are connected perpendicularly.
They have three different normals per vertex. On the other hand, smooth objects such as game
characters have one normal per vertex.

Lighting 3D Objects

301

Sample Program (LightedCube.js)
The sample program is shown in Listing 8.1. It is based on ColoredCube from the previous
chapter, so the basic processing flow of this program is the same as ColoredCube.
As you can see from Listing 8.1, the vertex shader has been significantly modified so that
it calculates Equation 8.4. In addition, the normal data is added in initVertexBuffers()
defined at line 89, so that they can be passed to the variable a_Normal. The fragment
shader is the same as in ColoredCube, and unmodified. It is reproduced so that you can
see that no fragment processing is needed.
Listing 8.1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22

LightedCube.js

// LightedCube.js
// Vertex shader program
var VSHADER_SOURCE =
'attribute vec4 a_Position;\n' +
'attribute vec4 a_Color;\n' +
'attribute vec4 a_Normal;\n' +
// Normal
'uniform mat4 u_MvpMatrix;\n' +
'uniform vec3 u_LightColor;\n' +
// Light color
'uniform vec3 u_LightDirection;\n' + // world coordinate, normalized
'varying vec4 v_Color;\n' +
'void main() {\n' +
' gl_Position = u_MvpMatrix * a_Position ;\n' +
// Make the length of the normal 1.0
' vec3 normal = normalize(vec3(a_Normal));\n' +
// Dot product of light direction and orientation of a surface
' float nDotL = max(dot(u_LightDirection, normal), 0.0);\n' +
// Calculate the color due to diffuse reflection
' vec3 diffuse = u_LightColor * vec3(a_Color) * nDotL;\n' +
' v_Color = vec4(diffuse, a_Color.a);\n' +
'}\n';

// Fragment shader program
...
28
'void main() {\n' +
29
' gl_FragColor = v_Color;\n' +
30
'}\n';
31
32 function main() {
...
49
// Set the vertex coordinates, the color, and the normal
50
var n = initVertexBuffers(gl);
...

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61
62
63
69
70
71
72
73
74
75
76
77
78
79
80
81

var u_MvpMatrix = gl.getUniformLocation(gl.program, 'u_MvpMatrix');
var u_LightColor = gl.getUniformLocation(gl.program, 'u_LightColor');
var u_LightDirection = gl.getUniformLocation(gl.program, 'u_LightDirection');
...
// Set the light color (white)
gl.uniform3f(u_LightColor, 1.0, 1.0, 1.0);
// Set the light direction (in the world coordinate)
var lightDirection = new Vector3([0.5, 3.0, 4.0]);
lightDirection.normalize();
// Normalize
gl.uniform3fv(u_LightDirection, lightDirection.elements);

// Calculate the view projection matrix
var mvpMatrix = new Matrix4();
// Model view projection matrix
mvpMatrix.setPerspective(30, canvas.width/canvas.height, 1, 100);
mvpMatrix.lookAt(3, 3, 7, 0, 0, 0, 0, 1, 0);
// Pass the model view projection matrix to the variable u_MvpMatrix
gl.uniformMatrix4fv(u_MvpMatrix, false, mvpMatrix.elements);
...
86
gl.drawElements(gl.TRIANGLES, n, gl.UNSIGNED_BYTE, 0);// Draw a cube
87 }
88
89 function initVertexBuffers(gl) {
...
98
var vertices = new Float32Array([ // Vertices
99
1.0, 1.0, 1.0, -1.0, 1.0, 1.0, -1.0,-1.0, 1.0,
1.0,-1.0, 1.0,
100
1.0, 1.0, 1.0,
1.0,-1.0, 1.0,
1.0,-1.0,-1.0,
1.0, 1.0,-1.0,
...
104
1.0,-1.0,-1.0, -1.0,-1.0,-1.0, -1.0, 1.0,-1.0,
1.0, 1.0,-1.0
105
]);
...
117
118
var normals = new Float32Array([ // Normals
119
0.0, 0.0, 1.0,
0.0, 0.0, 1.0,
0.0, 0.0, 1.0,
0.0, 0.0, 1.0,
120
1.0, 0.0, 0.0,
1.0, 0.0, 0.0,
1.0, 0.0, 0.0,
1.0, 0.0, 0.0,
...
124
0.0, 0.0,-1.0,
0.0, 0.0,-1.0,
0.0, 0.0,-1.0,
0.0, 0.0,-1.0
125
]);
...
if(!initArrayBuffer(gl,'a_Normal', normals, 3, gl.FLOAT)) return -1;
...
154
return indices.length;
155 }
140

Lighting 3D Objects

303

As a reminder, here is the calculation that the vertex shader performs (Equation 8.4):
〈 surface color by diffuse reflection 〉 =
〈light color 〉 × 〈base color of surface〉 ×

( 〈light

direction 〉 i 〈orientation of a surface〉 )

You can see that four pieces of information are needed to calculate this equation: (1) light
color, (2) a surface base color, (3) light direction, and (4) surface orientation. In addition,
 and  must be normalized (1.0 in length).
Processing in the Vertex Shader
From the four pieces of information necessary for Equation 8.4, the base color of a
surface is passed as a_Color at line 5 in the following code, and the surface orientation is
passed as a_Normal at line 6. The light color is passed using u_LightColor at line 8, and
the light direction is passed as u_LightDirection at line 9. You should note that only
u_LightDirection is passed in the world coordinate9 system and has been normalized in
the JavaScript code for ease of handling. This avoids the overhead of normalizing it every
time it’s used in the vertex shader:
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

'attribute vec4 a_Position;\n' +
'attribute vec4 a_Color;\n' +
<-(2) surface base color
'attribute vec4 a_Normal;\n' +
// Normal <-(4) surface orientation
'uniform mat4 u_MvpMatrix;\n' +
'uniform vec3 u_LightColor;\n' +
// Light color
<-(1)
'uniform vec3 u_LightDirection;\n' + // world coordinate,normalized <-(3)
'varying vec4 v_Color;\n' +
'void main() {\n' +
' gl_Position = u_MvpMatrix * a_Position ;\n' +
// Make the length of the normal 1.0
' vec3 normal = normalize(vec3(a_Normal));\n' +
// Dot product of light direction and orientation of a surface
' float nDotL = max(dot(u_LightDirection, normal), 0.0);\n' +
// Calculate the color due to diffuse reflection
' vec3 diffuse = u_LightColor * vec3(a_Color) * nDotL;\n' +
' v_Color = vec4(diffuse, a_Color.a);\n' +
'}\n';

Once the necessary information is available, you can carry out the calculation. First, the
vertex shader normalizes the vector at line 14. Technically, because the normal used in
this sample program is 1.0 in length, this process is not necessary. However, it is good
practice, so it is performed here:
9

In this book, the light effect with shading is calculated in the world coordinate system (see Appendix
G, “World Coordinate System Versus Local Coordinate System”) because it is simpler to program and
more intuitive with respect to the light direction. It is also safe to calculate it in the view coordinate
system but more complex.

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14

'

vec3 normal = normalize(vec3(a_Normal));\n' +

Although a_Normal is of type vec4, a normal represents a direction and uses only the x, y,
and z components. So you extract these components with .xyz and then normalize. If you
pass the normal using a type vec3, this process is not necessary. However, it is passed as
a type vec4 in this code because a vec4 will be needed when we extend the code for the
next example. We will explain the details in a later sample program. As you can see, GLSL
ES provides normalize(), a built-in function to normalize a vector specified as its argument. In the program, the normalized normal is stored in the variable normal for use later.
Next, you need to calculate the dot product 〈light direction 〉 i 〈 surface orientation 〉 from
Equation 8.4. The light direction is stored in u_LightDirection. Because it is already
normalized, you can use it as is. The orientation of the surface is the normal that was
normalized at line 14. The dot product “·” can then be calculated using the built-in
function dot(), which again is provided by GLSL ES and returns the dot product of the
two vectors specified as its arguments. That is, calling dot(u_LightDirection, normal)
performs 〈light direction 〉 i 〈 surface orientation 〉 . This calculation is performed at line 16.
16

'

float nDotL = max(dot(u_LightDirection, normal), 0.0);\n' +

Once the dot product is calculated, if the result is positive, it is assigned to nDotL. If it is
negative then 0.0 is assigned. The function max() used here is a GLSL ES built-in function
that returns the greater value from its two arguments.
A negative dot product means that θ in cos θ is more than 90 degrees. Because θ is the
angle between the light direction and the surface orientation, a value of θ greater than 90
degrees means that light hits the surface on its back face (see Figure 8.11). This is the same
as no light hitting the front face, so 0.0 is assigned to nDotL.
orientation of
the surface

θ > 90

diffuse reflection
light
direction
light
source

Figure 8.11

A normal and light in case θ is greater than 90 degrees

Now that the preparation is completed, you can calculate Equation 8.4. This is performed
at line 18, which is a direct implementation of Equation 8.4. a_Color, which is of type
vec4 and holds the RGBA values, is converted to a vec3 (.rgb) because its transparency
(alpha value) is not used in lighting.
Lighting 3D Objects

305

In fact, transparency of an object’s surface has a significant effect on the color of the
surface. However, because the calculation of the light passing through an object is complicated, we ignore transparency and don’t use the alpha value in this program:
18

'

vec3 diffuse = u_LightColor * vec3(a_Color) * nDotL;\n' +

Once calculated, the result, diffuse, is assigned to the varying variable v_Color at line 19.
Because v_Color is of type vec4, diffuse is also converted to vec4 with 1.0:
19

'

v_Color = vec4(diffuse, 1.0);\n' +

The result of the processing steps above is that a color, depending on the direction of
the vertex’s normal, is calculated, passed to the fragment shader, and assigned to gl_
FragColor. In this case, because you use a directional light, vertices that make up the same
surface are the same color, so each surface will be a solid color.
That completes the vertex shader code. Let’s now take a look at how the JavaScript
program passes the data needed for Equation 8.4 to the vertex shader.
Processing in the JavaScript Program
The light color (u_LightColor) and the light direction (u_LightDirection) are passed to
the vertex shader from the JavaScript program. Because the light color is white (1.0, 1.0,
1.0), it is simply written to u_LightColor using gl.uniform3f():
69
70

// set the Light color (white)
gl.uniform3f(u_LightColor, 1.0, 1.0, 1.0);

The next step is to set up the light direction, which must be passed after normalization, as
discussed before. You can normalize it with the normalize() function for Vector3 objects
that is provided in cuon-matrix.js. Usage is simple: Create the Vector3 object that specifies the vector you want to normalize as its argument (line 72), and invoke the normalize() method on the object. Note that the notation in JavaScript is different from that of
GLSL ES:
71
72
73
74

// Set the light direction (in the world coordinate)
var lightDirection = new Vector3([0.5, 3.0, 4.0]);
lightDirection.normalize();
// Normalize
gl.uniform3fv(u_LightDirection, lightDirection.elements);

The result is stored in the elements property of the object in an array of type
Float32Array and then assigned to u_LightDirection using gl.uniform3fv() (line 74).
Finally, the normal data is written in initVertexBuffers(), defined at line 89. Actual
normal data is stored in the array normals at line 118 per vertex along with the color
data, as in ColoredCube.js. Data is assigned to a_Normal in the vertex shader by invoking
initArrayBuffer() at line 140:
140 if(!initArrayBuffer(gl, 'a_Normal', normals, 3, gl.FLOAT)) return -1;

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initArrayBuffer(), which was also used in ColoredCube, assigns the array specified by
the third argument (normals) to the attribute variable that has the name specified by the
second argument (a_Normal).

Add Shading Due to Ambient Light
Although at this stage you have successfully added lighting to the scene, as you can see
from Figure 8.9, when you run LightedCube, the cube is a little different from the box in
the real world. In particular, the surface on the opposite side of the light source appears
almost black and not clearly visible. You can see this problem more clearly if you animate
the cube. Try the sample program LightedCube_animation (see Figure 8.12) to see the
problem more clearly.

Figure 8.12

The result of LightedCube_animation

Although the scene is correctly lit as the result of Equation 8.4, our real-world experiences
tells us that something isn’t right. It is unusual to see such a sharp effect because, in the
real world, surfaces such as the back face of the cube are also lit by diffuse or reflected
light. The ambient light described in the previous section represents this indirect light
and can be used to make the scene more lifelike. Let’s add that to the scene and see if the
effect is more realistic. Because ambient light models the light that hits an object from all
directions with constant intensity, the surface color due to the reflection is determined
only by the light color and the base color of the surface. The formula that calculates this
was shown as Equation 8.2. Let’s see it again:
〈 surface color by ambient reflection 〉 =
〈light color 〉 × 〈base color of surface〉

Lighting 3D Objects

307

Let’s try to add the color due to ambient light described by this formula to the sample
program LightedCube. To do this, use Equation 8.3 shown here:
〈 surface color by diffuse and ambient reflection 〉 =
〈 surface color by diffuse reflection 〉 + 〈 surface color by ambient reflection 〉
Ambient light is weak because it is the light reflected by other objects like the walls. For
example, if the ambient light color is (0.2, 0.2, 0.2) and the base color of a surface is red,
or (1.0, 0.0, 0.0), then, from Equation 8.2, the surface color due to the ambient light is
(0.2, 0.0, 0.0). For example, if there is a white box in a blue room—that is, the base color
of the surface is (1.0, 1.0, 1.0) and the ambient light is (0.0, 0.0, 0.2)—the color becomes
slightly blue (0.0, 0.0, 0.2).
Let’s implement the effect of ambient reflection in the sample program LightedCube_
ambient, which results in the cube shown in Figure 8.13. You can see that the surface that
the light does not directly hit is now also slightly colored and more closely resembles the
cube in the real world.

Figure 8.13

LightedCube_ambient

Sample Program (LightedCube_ambient.js)
Listing 8.2 illustrates the sample program. Because it is almost the same as LightedCube,
only the modified parts are shown.
Listing 8.2

LightedCube_ambient.js

1 // LightedCube_ambient.js
2 // Vertex shader program
...

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8
9
10
11
12

'uniform vec3 u_LightColor;\n' +
// Light color
'uniform vec3 u_LightDirection;\n' + // World coordinate, normalized
'uniform vec3 u_AmbientLight;\n' +
// Color of an ambient light
'varying vec4 v_Color;\n' +
'void main() {\n' +
...

16
17
18
19
20
21
22
23
24
36
64
68
80
81
95

// The dot product of the light direction and the normal
float nDotL = max(dot(lightDirection, normal), 0.0);\n' +
// Calculate the color due to diffuse reflection
' vec3 diffuse = u_LightColor * a_Color.rgb * nDotL;\n' +
// Calculate the color due to ambient reflection
' vec3 ambient = u_AmbientLight * a_Color.rgb;\n' +
// Add surface colors due to diffuse and ambient reflection
' v_Color = vec4(diffuse + ambient, a_Color.a);\n' +
'}\n';
...
function main() {
...
// Get the storage locations of uniform variables and so on
...
var u_AmbientLight = gl.getUniformLocation(gl.program, 'u_AmbientLight');
...
// Set the ambient light
gl.uniform3f(u_AmbientLight, 0.2, 0.2, 0.2);
...
}
'

u_AmbientLight at line 10 is added to the vertex shader to pass in the color of ambient
light. After Equation 8.2 is calculated using it and the base color of the surface (a_Color),
the result is stored in the variable ambient (line 21). Now that both diffuse and ambient
are determined, the surface color is calculated at line 23 using Equation 8.3. The result is
passed to v_Color, just like in LightedCube, and the surface is painted with this color.

As you can see, this program simply adds ambient at line 23, causing the whole cube to
become brighter. This implements the effect of the ambient light hitting an object equally
from all directions.
The examples so far have been able to handle static objects. However, because objects are
likely to move within a scene, or the viewpoint changes, you have to be able to handle
such transformations. As you will recall from Chapter 4, “More Transformations and Basic
Animation,” an object can be translated, scaled, or rotated using coordinate transformations. These transformations may also change the normal direction and require a recalculation of lighting as the scene changes. Let’s take a look at how to achieve that.

Lighting 3D Objects

309

Lighting the Translated-Rotated Object
The program LightedTranslatedRotatedCube uses a directional light source to light a cube
that is rotated 90 degrees clockwise around the z-axis and translated 0.9 units in the y-axis
direction. A part from a directional light as described in the previous section, the sample,
LightedCube_ambient, uses diffuse reflection and ambient reflection and rotates and translates the cube. The result is shown in Figure 8.14.

Figure 8.14

LightedTranslatedRotatedCube

You saw in the previous section that the normal direction may change when coordinate
transformations are applied. Figure 8.15 shows some examples of that. The leftmost figure
in Figure 8.15 shows the cube used in this sample program looking along the negative
direction of the z-axis. The only normal (1, 0, 0), which is toward the positive direction of
the x-axis, is shown. Let’s perform some coordinate transforms on this figure, which are
the three figures on the right.
y

the normal direction
(1 0 0)

the normal direction
(1 1 0)

the normal direction
(1, 0.5, 0)

x
the normal
direction
(1 0 0)
the original normal

Figure 8.15

310

(1) translation
(translate along the y-axis)

(2) rotation
(rotate 45 degrees)

(3) scaling
(scale by 2 along
the y-axis)

The changes of the normal direction due to coordinate transformations

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Lighting Objects

You can see the following from Figure 8.15:

• The normal direction is not changed by a translation because the orientation of the
object does not change.

• The normal direction is changed by a rotation according to the orientation of the
object.

• Scaling has a more complicated effect on the normal. As you can see, the object in
the rightmost figure is rotated i and then scaled two times only in the y-axis. In
this case, the normal direction is changed because the orientation of the surface
changes. On the other hand, if an object is scaled equally in all axes the normal
direction is not changed. Finally, even if an object is scaled unequally, the normal
direction may not change. For example, when the leftmost figure (the original
normal) is scaled two times only in the y-axis direction, the normal direction does
not change.
Obviously, the calculation of the normal under various transformations is complex, particularly when dealing with scaling. However, a mathematical technique can help.

The Magic Matrix: Inverse Transpose Matrix
As described in Chapter 4, the matrix that performs a coordinate transformation on an
object is called a model matrix. The normal direction can be calculated by multiplying the
normal by the inverse transpose matrix of a model matrix. The inverse transpose matrix
is the matrix that transposes the inverse of a matrix.
The inverse of the matrix M is the matrix R, where both R*M and M*R become the identity matrix. The term transpose means the operation that exchanges rows and columns of
a matrix. The details of this are explained in Appendix E, “The Inverse Transpose Matrix.”
For our purposes, it can be summarized simply using the following rule:
Rule: You can calculate the normal direction if you multiply the normal by the
inverse transpose of the model matrix.
The inverse transpose matrix is calculated as follows:
1. Invert the original matrix.
2. Transpose the resulting matrix.
This can be carried out using convenient methods supported by the Matrix4 object (see
Table 8.1).

Lighting the Translated-Rotated Object

311

Table 8.1

Matrix4 Methods for an Inverse Transpose Matrix

Method

Description

Matrix4.setInverseOf(m)

Calculates the inverse of the matrix stored in m and stores the
result in the Matrix4 object, where m is a Matrix4 object

Matrix4.transpose()

Transposes the matrix stored in the Matrix4 object and writes
the result back into the Matrix4 object

Assuming that a model matrix is stored in modelMatrix, which is a Matrix4 object, the
following code snippet will get its inverse transpose matrix. The result is stored in the variable named normalMatrix, because it performs the coordinate transformation of a normal:
Matrix4 normalMatrix = new Matrix4();
// Calculate the model matrix
...
// Calculate the matrix to transform normal according to the model matrix
normalMatrix.setInverseOf(modelMatrix);
normalMatrix.transpose();

Now let’s see the program LightedTranslatedRotatedCube.js that lights the cube, which
is rotated 90 degrees clockwise around the z-axis and translated 0.9 along the y-axis, all
using directional light. You’ll use the cube that was transformed by the model matrix in
LightedCube_ambient from the previous section.

Sample Program (LightedTranslatedRotatedCube.js)
Listing 8.3 shows the sample program. The changes from LightedCube_ambient are that
u_NormalMatrix is added (line 8) to pass the matrix for coordinate transformation of the
normal to the vertex shader, and the normal is transformed at line 16 using this matrix.
u_NormalMatrix is calculated within the JavaScript.
Listing 8.3

LightedTranslatedRotatedCube.js

1 // LightedTranslatedRotatedCube.js
2 // Vertex shader program
3 var VSHADER_SOURCE =
...
6
'attribute vec4 a_Normal;\n' +
7
'uniform mat4 u_MvpMatrix;\n' +
8
'uniform mat4 u_NormalMatrix;\n'+
//
9
'uniform vec3 u_LightColor;\n' +
//
10
'uniform vec3 u_LightDirection;\n' + //
11
'uniform vec3 u_AmbientLight;\n' +
//
12
'varying vec4 v_Color;\n' +

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Transformation matrix of normal
Light color
World coordinate, normalized
Ambient light color

13
14
15
16
17
18
19
20
21
22
23
24
25
37
65
66
67
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102

'void main() {\n' +
' gl_Position = u_MvpMatrix * a_Position;\n' +
// Recalculate normal with normal matrix and make its length 1.0
' vec3 normal = normalize(vec3(u_NormalMatrix * a_Normal));\n' +
// The dot product of the light direction and the normal
' float nDotL = max(dot(u_LightDirection, normal), 0.0);\n' +
// Calculate the color due to diffuse reflection
' vec3 diffuse = u_LightColor * a_Color.rgb * nDotL;\n' +
// Calculate the color due to ambient reflection
' vec3 ambient = u_AmbientLight * a_Color.rgb;\n' +
// Add the surface colors due to diffuse and ambient reflection
' v_Color = vec4(diffuse + ambient, a_Color.a);\n' +
'}\n';
...
function main() {
...
// Get the storage locations of uniform variables and so on
var u_MvpMatrix = gl.getUniformLocation(gl.program, 'u_MvpMatrix');
var u_NormalMatrix = gl.getUniformLocation(gl.program, 'u_NormalMatrix');
...
var modelMatrix = new Matrix4(); // Model matrix
var mvpMatrix = new Matrix4();
// Model view projection matrix
var normalMatrix = new Matrix4(); // Transformation matrix for normal
// Calculate the model matrix
modelMatrix.setTranslate(0, 1, 0); // Translate to y-axis direction
modelMatrix.rotate(90, 0, 0, 1);
// Rotate around the z-axis
// Calculate the view projection matrix
mvpMatrix.setPerspective(30, canvas.width/canvas.height, 1, 100);
mvpMatrix.lookAt(-7, 2.5, 6, 0, 0, 0, 0, 1, 0);
mvpMatrix.multiply(modelMatrix);
// Pass the model view projection matrix to u_MvpMatrix
gl.uniformMatrix4fv(u_MvpMatrix, false, mvpMatrix.elements);
// Calculate matrix to transform normal based on the model matrix
normalMatrix.setInverseOf(modelMatrix);
normalMatrix.transpose();
// Pass the transformation matrix for normal to u_NormalMatrix

103

gl.uniformMatrix4fv(u_NormalMatrix, false, normalMatrix.elements);
...
110 }

Lighting the Translated-Rotated Object

313

The processing in the vertex shader is almost the same as in LightedCube_ambient. The
difference, in line with the preceding rule, is that you multiply a_Normal by the inverse
transpose of the model matrix at line 16 instead of using it as-is:
15
16

// Recalculate normal with normal matrix and make its length 1.0
vec3 normal = normalize(vec3(u_NormalMatrix * a_Normal));\n' +

'

Because you passed a_Normal as type vec4, you can multiply it by u_NormalMatrix, which
is of type mat4. You only need the x, y, and z components of the result of the multiplication, so the result is converted into type vec3 with vec3(). It is also possible to use .xyz
as before, or write (u_NormalMatrix * a_Normal).xyz. However, vec3() is used here for
simplicity. Now that you understand how the shader calculates the normal direction
resulting from the rotation and translation of the object, let’s move on to the explanation
of the JavaScript program. The key point here is the calculation of the matrix that will be
passed to u_NormalMatrix in the vertex shader.
u_NormalMatrix is the inverse transpose of the model matrix, so the model matrix is first

calculated at lines 90 and 91. Because this program rotates an object around the z-axis
and translates it in the y-axis direction, you can use the setTranslate() and rotate()
methods of a Matrix4 object as described in Chapter 4. It is at lines 100 and 101 that the
inverse transpose matrix is actually calculated. It is passed to u_NormalMatrix in the vertex
shader at line 103, in the same way as mvpMatrix at line 97. The second argument of gl.
uniformMatrix4f() specifies whether to transpose the matrix (Chapter 3):
99
100
101
102
103

// Calculate matrix to transform normal based on the model matrix
normalMatrix.setInverseOf(modelMatrix);
normalMatrix.transpose();
// Pass the normal transformation matrix to u_NormalMatrix
gl.uniformMatrix4fv(u_NormalMatrix, false, normalMatrix.elements);

When run, the output is similar to Figure 8.14. As you can see, the shading is the same as
LightedCube_ambient with the cube translated in the y-axis direction. That is because (1)
the translation doesn’t change the normal direction, (2) neither does the rotation by 90
degrees, because the rotation simply switches the surfaces of the cube, (3) the light direction of the directional light does not change regardless of the position of the object, and
(4) diffuse reflection reflects the light in all directions with equal intensity.
You now have a good understanding of the basics of how to implement light and shade in
3D graphics. Let’s build on this by exploring another type of light source: the point light.

Using a Point Light Object
In contrast to a directional light, the direction of the light from a point light source differs
at each position in the 3D scene (see Figure 8.16). So, when calculating shading, you need
to calculate the light direction at the specific position on the surface where the light hits.

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a point light

α

Figure 8.16

β

The direction of a point light varies by position

In the previous sample programs, you calculated the color at each vertex by passing the
normal and the light direction for each vertex. You will use the same approach here, but
because the light direction changes, you need to pass the position of the light source and
then calculate the light direction at each vertex position.
Here, you construct the sample program PointLightedCube that displays a red cube lit
with white light from a point light source. We again use diffuse reflection and ambient
reflection. The result is shown in Figure 8.17, which is a version of LightedCube_ambient
from the previous section but now lit with a point light.

Figure 8.17

PointLightedCube

Sample Program (PointLightedCube.js)
Listing 8.4 shows the sample program in which only the vertex shader is changed from
LightedCube_ambient. The variable u_ModelMatrix for passing the model matrix and the
variable u_LightPosition representing the light position are added. Note that because you

Using a Point Light Object

315

use a point light in this program, you will use the light position instead of the light direction. Also, to make the effect easier to see, we have enlarged the cube.
Listing 8.4

PointLightedCube.js

1 // PointLightedCube.js
2 // Vertex shader program
3 var VSHADER_SOURCE =
4
'attribute vec4 a_Position;\n' +
...
8
'uniform mat4 u_ModelMatrix;\n' + // Model matrix
9
'uniform mat4 u_NormalMatrix;\n' + // Transformation matrix of normal
10
'uniform vec3 u_LightColor;\n' +
// Light color
11
'uniform vec3 u_LightPosition;\n' + // Position of the light source (in the
➥world coordinate system)
12
'uniform vec3 u_AmbientLight;\n' +
// Ambient light color
13
'varying vec4 v_Color;\n' +
14
'void main() {\n' +
15
' gl_Position = u_MvpMatrix * a_Position;\n' +
16
// Recalculate normal with normal matrix and make its length 1.0
17
' vec3 normal = normalize(vec3(u_NormalMatrix * a_Normal));\n' +
18
// Calculate the world coordinate of the vertex
19
' vec4 vertexPosition = u_ModelMatrix * a_Position;\n' +
20
// Calculate the light direction and make it 1.0 in length
21
' vec3 lightDirection = normalize(u_LightPosition – vec3(vertexPosition));\n' +
22
// The dot product of the light direction and the normal
23
' float nDotL = max(dot( lightDirection, normal), 0.0);\n' +
24
// Calculate the color due to diffuse reflection
25
' vec3 diffuse = u_LightColor * a_Color.rgb * nDotL;\n' +
26
// Calculate the color due to ambient reflection
27
' vec3 ambient = u_AmbientLight * a_Color.rgb;\n' +
28
// Add surface colors due to diffuse and ambient reflection
29
' v_Color = vec4(diffuse + ambient, a_Color.a);\n' +
30
'}\n';
...
42 function main() {
...
70
// Get the storage locations of uniform variables and so on
71
var u_ModelMatrix = gl.getUniformLocation(gl.program, 'u_ModelMatrix');
...
74
var u_LightColor = gl.getUniformLocation(gl.program,'u_LightColor');
75
var u_LightPosition = gl.getUniformLocation(gl.program, 'u_LightPosition');
...
82
// Set the light color (white)

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83
84
85
89
90
91
92
93
94
95
96

gl.uniform3f(u_LightColor, 1.0, 1.0, 1.0);
// Set the position of the light source (in the world coordinate)
gl.uniform3f(u_LightPosition, 0.0, 3.0, 4.0);
...
var modelMatrix = new Matrix4(); // Model matrix
var mvpMatrix = new Matrix4();
// Model view projection matrix
var normalMatrix = new Matrix4(); // Transformation matrix for normal
// Calculate the model matrix
modelMatrix.setRotate(90, 0, 1, 0); // Rotate around the y-axis
// Pass the model matrix to u_ModelMatrix
gl.uniformMatrix4fv(u_ModelMatrix, false, modelMatrix.elements);
...

The key differences in the processing within the vertex shader are at line 19 and 21. At
line 19, you transform the vertex coordinates into world coordinates in order to calculate
the light direction at the vertex coordinates. Because a point light emits light in all directions from its position, the light direction at a vertex is the result of subtracting the vertex
position from the light source position. Because the light position is passed to the variable
u_LightPosition using world coordinates at line 11, you also have to convert the vertex
coordinates into world coordinates to calculate the light direction. The light direction
is then calculated at line 21. Note that it is normalized with normalize() so that it will
be 1.0 in length. Using the resulting light direction (lightDirection), the dot product is
calculated at line 23 and then the surface color at each vertex is calculated based on this
light direction.
If you run this program, you will see a more realistic result, as shown in Figure 8.17.
Although this result is more realistic, a closer look reveals an artifact: There are unnatural
lines of shade on the cube’s surface (see Figure 8.18). You can see this more easily if the
cube rotates as it does when you load PointLightedCube_animation.

Using a Point Light Object

317

Figure 8.18

The unnatural appearance when processing the point light at each vertex

This comes about because of the interpolation process discussed in Chapter 5, “Using
Colors and Texture Images.” As you will remember, the WebGL system interpolates the
colors between vertices based on the colors you supply at the vertices. However, because
the direction of light from a point light source varies by position to shade naturally, you
have to calculate the color at every position the light hits instead of just at each vertex.
You can see this problem more clearly using a sphere illuminated by a point light, as
shown in Figure 8.19.

per-vertex
calculation

Figure 8.19

per-position
calculation

The spheres illuminated by a point light

As you can see, the border between the brighter parts and darker parts is unnatural in the
left figure. If the effect is hard to see on the page, the left figure is PointLightedSphere,
and the right is PointLightedSphere_perFragment. We will describe how to draw them
correctly in the next section.
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More Realistic Shading: Calculating the Color per Fragment
At first glance, it may seem daunting to have to calculate the color at every position on a
cube surface where the light hits. However, essentially it means calculating the color per
fragment, so the power of the fragment shader can now be used.
This sample program you will use is PointLightedCube_perFragment, and its result is
shown in Figure 8.20.

Figure 8.20

PointLightedCube_perFragment

Sample Program (PointLightedCube_perFragment.js)
The sample program, which is based on PointLightedCube.js, is shown in Listing 8.5.
Only the shader code has been modified and, as you can see, there is less processing in the
vertex shader and more processing in the fragment shader.
Listing 8.5

PointLightedCube_perFragment.js

1 // PointLightedCube_perFragment.js
2 // Vertex shader program
3 var VSHADER_SOURCE =
4
'attribute vec4 a_Position;\n' +
...
8
'uniform mat4 u_ModelMatrix;\n' + // Model matrix
9
'uniform mat4 u_NormalMatrix;\n' + // Transformation matrix of normal
10
'varying vec4 v_Color;\n' +
11
'varying vec3 v_Normal;\n' +
12
'varying vec3 v_Position;\n' +
13
'void main() {\n' +

Using a Point Light Object

319

14
' gl_Position = u_MvpMatrix * a_Position;\n' +
15
// Calculate the vertex position in the world coordinate
16
' v_Position = vec3(u_ModelMatrix * a_Position);\n' +
17
' v_Normal = normalize(vec3(u_NormalMatrix * a_Normal));\n' +
18
' v_Color = a_Color;\n' +
19
'}\n';
20
21 // Fragment shader program
22 var FSHADER_SOURCE =
...
26
'uniform vec3 u_LightColor;\n' +
// Light color
27
'uniform vec3 u_LightPosition;\n' + // Position of the light source
28
'uniform vec3 u_AmbientLight;\n' +
// Ambient light color
29
'varying vec3 v_Normal;\n' +
30
'varying vec3 v_Position;\n' +
31
'varying vec4 v_Color;\n' +
32
'void main() {\n' +
33
// Normalize normal because it's interpolated and not 1.0 (length)
34
' vec3 normal = normalize(v_Normal);\n' +
35
// Calculate the light direction and make it 1.0 in length
36
' vec3 lightDirection = normalize(u_LightPosition - v_Position);\n' +
37
// The dot product of the light direction and the normal
38
' float nDotL = max(dot( lightDirection, normal), 0.0);\n' +
39
// Calculate the final color from diffuse and ambient reflection
40
' vec3 diffuse = u_LightColor * v_Color.rgb * nDotL;\n' +
41
' vec3 ambient = u_AmbientLight * v_Color.rgb;\n' +
42
' gl_FragColor = vec4(diffuse + ambient, v_Color.a);\n' +
43
'}\n';

To calculate the color per fragment when light hits, you need (1) the position of the fragment in the world coordinate system and (2) the normal direction at the fragment position. You can utilize interpolation (Chapter 5) to obtain these values per fragment by just
calculating them per vertex in the vertex shader and passing them via varying variables to
the fragment shader.
These calculations are performed at lines 16 and 17, respectively, in the vertex shader. At
line 16, the vertex position in world coordinates is calculated by multiplying each vertex
coordinate by the model matrix. After assigning the vertex position to the varying variable v_Position, it will be interpolated between vertices and passed to the corresponding
variable (v_Position) in the fragment shader as the world coordinate of the fragment. The
normal calculation at line 17 is carried out for the same purpose.10 By assigning the result
to v_Normal, it is also interpolated and passed to the corresponding variable (v_Normal) in
the fragment shader as the normal of the fragment.

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Processing in the fragment shader is the same as that in the vertex shader of
PointLightedCube.js. First, at line 34, the interpolated normal passed from the vertex
shader is normalized. Its length may not be 1.0 anymore because of the interpolation.
Next, at line 36, the light direction is calculated and normalized. Using these results, the
dot product of the light direction and the normal is calculated at line 38. The colors due
to the diffuse reflection and ambient reflection are calculated at lines 40 and 41 and added
to get the fragment color, which is assigned to gl_FragColor at line 42.
If you have more than one light source, after calculating the color due to diffuse reflection and ambient reflection for each light source, you can obtain the final fragment color
by adding all the colors. In other words, you only have to calculate Equation 8.3 as many
times as the number of light sources.

Summary
This chapter explored how to light a 3D scene, the different types of light used, and how
light is reflected and diffused through the scene. Using this knowledge, you then implemented the effects of different light sources to illuminate a 3D object and examined
various shading techniques to improve the realism of the objects. As you have seen, a
mastery of lighting is essential to adding realism to 3D scenes, which can appear flat and
uninteresting if they’re not correctly lit.

10

In this sample program, this normalization is not necessary because all normals are passed to
a_Normal with a length of 1.0. However, we normalize them here as good programming practice so
the code is more generic.

Summary

321

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Chapter 9

Hierarchical Objects

This chapter is the final one that describes the core features and how to program
with WebGL. Once you’ve read it, you will have mastered the basics of WebGL
and will have enough knowledge to create realistic and interactive 3D scenes.
This chapter focuses on hierarchical objects, which are important because
they allow you to progress beyond single objects like cubes or blocks to more
complex objects that you can use for game characters, robots, and even humans.
The following key points are discussed in this chapter:

• Modeling complex connected structures such as a robot arm using a hierarchical structure.

• Drawing and manipulating hierarchical objects made up of multiple
simpler objects.

• Combining model and rotation matrices to mimic joints such as elbow or
wrist joints

• Internally implementing initShaders(), which you’ve used but not examined so far.
By the end of this chapter, you will have all the knowledge you need to create
compelling 3D scenes populated by both simple and complex 3D objects.

Drawing and Manipulating Objects Composed of
Other Objects
Until now, we have described how to translate and rotate a single object, such as a twodimensional triangle or a three-dimensional cube. But many of the objects in 3D graphics,
game characters, robots, and so on, consist of more than one object (or segment). For a
simple example, a robot arm is shown in Figure 9.1. As you can see, this consists of multiple boxes. The program name is MultiJointModel. First, let’s load the program and experiment by pressing the arrow, x, z, c, and v keys to understand what you will construct in
the following sections.

Figure 9.1

A robot arm consisting of multiple objects

One of the key issues when drawing an object consisting of multiple objects (segments)
is that you have to program to avoid conflicts when the segments move. This section will
explore this issue by describing how to draw and manipulate a robot arm that consists
of multiple segments. First, let’s consider the structure of the human body from the
shoulder to the fingertips to understand how to model our robot arm. An arm consists of
multiple segments, such as the upper arm, lower arm, palm, and fingers, each of which is
connected by a joint, as shown on the left of Figure 9.2.

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s h o u ld e r jo in t

shoulder joint

upper arm

u p p e r a rm
elbow joint

lower arm

lower arm
w r is t jo in t

palm
finger1
finger2

p a lm
finger1 finger2

rotate the shoulder joint

Figure 9.2

rotate the wrist joint

The structure and movement from the arm to the fingers

Each segment moves around a joint as follows:

• When you move the upper arm by rotating around the shoulder joint, depending
on the upper arm movement, the lower arm, palm, and fingers move (the middle of
Figure 9.2) accordingly.

• When you move the lower arm using an elbow joint, the palm and fingers move but
the upper arm does not.

• When you move the palm using the wrist joint, both palm and fingers move but the
upper and lower arm do not (the right of Figure 9.2).

• When you move fingers, the upper arm, lower arm, and palm do not move.
To summarize, when you move a segment, the segments located below it move, while the
segments located above are not affected. In addition, all movement, including twisting, is
actually rotation around a joint.

Hierarchical Structure
The typical method used to draw and manipulate the object with such features is to draw
each part object (such as a box) in the order of the object’s hierarchical structure from
upper to lower, applying each model matrix (rotation matrix) at every joint. For example,
in Figure 9.2, shoulder, elbow, wrist, and finger joints all have respective rotation matrices.
It is important to note that, unlike humans or robots, segments in 3D graphics are not
physically joined. So if you inadvertently rotate the object corresponding to an upper arm
at the shoulder joint, the lower parts would be left behind. When you rotate the shoulder
joint, you should explicitly make the lower parts follow the movement. To do this, you
need to rotate the lower elbow and wrist joints through the same angle that you rotate the
shoulder joint.

Drawing and Manipulating Objects Composed of Other Objects

325

It is straightforward to program so that the rotation of one segment propagates to the
lower segments and simply requires that you use the same model matrix for the rotation
of the lower segments. For example, when you rotate a shoulder joint through 30 degrees
using one model matrix, you can draw the lower elbow and wrist joints rotated through
30 degrees using the same model matrix (see Figure 9.3). Thus, by changing only the angle
of the shoulder rotation, the lower segments are automatically rotated to follow the movement of the shoulder joint.

30 degrees
30 degrees

rotate the shoulder joint

Figure 9.3

The lower segments following the rotation of the upper segment

For more complex cases, such as when you want to rotate the elbow joint 10 degrees after
rotating the shoulder joint 30 degrees, you can rotate the elbow joint by using the model
matrix and rotating 10 degrees more than the shoulder-joint model matrix. This can be
calculated by multiplying the shoulder-joint model matrix by a 10-degree rotation matrix,
which we refer to as the “elbow-joint model matrix.” The parts below the elbow will
follow the movement of the elbow when drawn using this elbow-joint model matrix.
By programming in such a way, the upper segments are not affected by rotation of the
lower segments. Thus, the upper segments will not move no matter how much the lower
segments move.
Now that you have a good understanding of the principles involved when moving multisegment objects, let’s look at a sample program.

Single Joint Model
Let’s begin with a simple single joint model. You will construct the program JointModel
that draws a robot arm consisting of two parts that can be manipulated with the arrow
keys. The screen shot and the hierarchy structure are shown on the left and right of Figure
9.4, respectively. This robot arm consists of arm1 and arm2, which are joined by joint1.
You should imagine that the arm is raised above the shoulder and that arm1 is the upper
part and arm2 the lower part. When you add the hand later, it will become clearer.

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a rm 2

arm2
joint1
arm1

Figure 9.4

r o ta tin g a r o u n d th e z - a x is
a rm 1

r o ta tin g a r o u n d th e y - a x is

JointModel and the hierarchy structure used in the program

If you run the program, you will see that arm1 is rotated around the y-axis using the right
and left arrow keys, and joint1 is rotated around the z-axis with the up and down arrow
keys (Figure 9.5). When pressing the down arrow key, joint1 is rotated and arm2 leans
forward, as shown on the left of Figure 9.5. Then if you press the right arrow key, arm1 is
rotated, as shown on the right of Figure 9.5.

arm2

arm1

Figure 9.5

The display change when pressing the arrow keys in JointModel

As you can see, the movement of arm2 by rotation of joint1 does not affect arm1. In
contrast, arm2 is rotated if you rotate arm1.
Drawing and Manipulating Objects Composed of Other Objects

327

Sample Program (JointModel.js)
JointModel.js is shown in Listing 9.1. The actual vertex shader is a little complicated

because of the shading process and has been removed from the listing here to save space.
However, if you are interested in how the lessons learned in the earlier part of the chapter
are applied, please look at the full listing available by downloading the examples from the
book website. The lighting used is a directional light source and simplified diffuse reflection, which makes the robot arm look more three-dimensional. However, as you can see,
there are no special lighting calculations needed for this joint model, and all the code
required to draw and manipulate the joint model is in the JavaScript program.
Listing 9.1

JointModel.js

1 // JointModel.js
2 // Vertex shader program
3 var VSHADER_SOURCE =
4
'attribute vec4 a_Position;\n' +
5
'attribute vec4 a_Normal;\n' +
6
'uniform mat4 u_MvpMatrix;\n' +
...
9
'void main() {\n' +
10
' gl_Position = u_MvpMatrix * a_Position;\n' +
11
// Shading calculation to make the arm look three-dimensional
...
17
'}\n';
...
29 function main() {
...
46
// Set the vertex coordinate.
47
var n = initVertexBuffers(gl);
...
57
// Get the storage locations of uniform variables
58
var u_MvpMatrix = gl.getUniformLocation(gl.program, 'u_MvpMatrix');
59
var u_NormalMatrix = gl.getUniformLocation(gl.program, 'u_NormalMatrix');
...
65
// Calculate the view projection matrix
66
var viewProjMatrix = new Matrix4();
67
viewProjMatrix.setPerspective(50.0, canvas.width / canvas.height, 1.0, 100.0);
68
viewProjMatrix.lookAt(20.0, 10.0, 30.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0);
69
70
// Register the event handler to be called when keys are pressed
71
document.onkeydown = function(ev){ keydown(ev, gl, n, viewProjMatrix,
➥u_MvpMatrix, u_NormalMatrix); };
72
// Draw robot arm
73
draw(gl, n, viewProjMatrix, u_MvpMatrix, u_NormalMatrix);
74 }

328

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Hierarchical Objects

75
76
77
78
79
80
81
82
83
84
85
86
87

var ANGLE_STEP = 3.0;
// The increments of rotation angle (degrees)
var g_arm1Angle = 90.0; // The rotation angle of arm1 (degrees)
var g_joint1Angle = 0.0; // The rotation angle of joint1 (degrees)
function keydown(ev, gl, n, viewProjMatrix, u_MvpMatrix, u_NormalMatrix) {
switch (ev.keyCode) {
case 38: // Up arrow key -> positive rotation of joint1 (z-axis)
if (g_joint1Angle < 135.0) g_joint1Angle += ANGLE_STEP;
break;
case 40: // Down arrow key -> negative rotation of joint1 (z-axis)
if (g_joint1Angle > -135.0) g_joint1Angle -= ANGLE_STEP;
break;
...
case 37: // Left arrow key -> negative rotation of arm1 (y-axis)
g_arm1Angle = (g_arm1Angle - ANGLE_STEP) % 360;
break;
default: return;
}
// Draw the robot arm
draw(gl, n, viewProjMatrix, u_MvpMatrix, u_NormalMatrix);
}

91
92
93
94
95
96
97
98
99
100 function initVertexBuffers(gl) {
101
// Vertex coordinates
...
148 }
...
174 // Coordinate transformation matrix
175 var g_modelMatrix = new Matrix4(), g_mvpMatrix = new Matrix4();
176
177 function draw(gl, n, viewProjMatrix, u_MvpMatrix, u_NormalMatrix) {
...
181
182
183
184
185
186
187
188
189
190
191
192 }

// Arm1
var arm1Length = 10.0; // Length of arm1
g_modelMatrix.setTranslate(0.0, -12.0, 0.0);
g_modelMatrix.rotate(g_arm1Angle, 0.0, 1.0, 0.0);
// Rotate y-axis
drawBox(gl, n, viewProjMatrix, u_MvpMatrix, u_NormalMatrix); // Draw
// Arm2
g_modelMatrix.translate(0.0, arm1Length, 0.0);
// Move to joint1
g_modelMatrix.rotate(g_joint1Angle, 0.0, 0.0, 1.0);// Rotate z-axis
g_modelMatrix.scale(1.3, 1.0, 1.3); // Make it a little thicker
drawBox(gl, n, viewProjMatrix, u_MvpMatrix, u_NormalMatrix); // Draw

Drawing and Manipulating Objects Composed of Other Objects

329

193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208

var g_normalMatrix = new Matrix4(); // Transformation matrix for normal
// Draw a cube
function drawBox(gl, n, viewProjMatrix, u_MvpMatrix, u_NormalMatrix) {
//Calculate the model view project matrix and pass it to u_MvpMatrix
g_mvpMatrix.set(viewProjMatrix);
g_mvpMatrix.multiply(g_modelMatrix);
gl.uniformMatrix4fv(u_MvpMatrix, false, g_mvpMatrix.elements);
// Calculate the normal transformation matrix and pass it to u_NormalMatrix
g_normalMatrix.setInverseOf(g_modelMatrix);
g_normalMatrix.transpose();
gl.uniformMatrix4fv(u_NormalMatrix, false, g_normalMatrix.elements);
// Draw
gl.drawElements(gl.TRIANGLES, n, gl.UNSIGNED_BYTE, 0);
}

The function main() from line 29 follows the same structure as before, with the first major
difference being the initVertexBuffers() function call at line 47. In initVertexBuffers(), the vertex data for arm1 and arm2 are written into the appropriate buffer objects.
Until now, you’ve been using cubes, with each side being 2.0 in length and the origin
at the center of the cube. Now, to better model the arm, you will use a cuboid like that
shown in the left side of Figure 9.6. The cuboid has its origin at the center of the bottom
surface and is 3.0 by 3.0 and 10.0 units in height. By setting the origin at the center of the
bottom surface, its rotation around the z-axis is the same as that of joint1 in Figure 9.5,
making it convenient to program. Both arm1 and arm2 are drawn using this cuboid.

y
y
v6(-1, 1, -1)
v5(1, 1, -1)
v1(-1, 1, 1)
v0(1, 1, 1)

10.0
v7(-1, -1, -1)

v4(1, -1, -1)
v2(-1, -1, 1)

z

v3(1, -1, 1)

x
z

Figure 9.6

330

3.0

3.0

A cuboid for drawing the robot arm

CHAPTER 9

Hierarchical Objects

the previous cube

x

From lines 66 to 68, a view projection matrix (viewProjMatrix) is calculated with the
specified viewing volume, the eye position, and the view direction.
Because the robot arm in this program is moved by using the arrow keys, the event
handler keydown() is registered at line 71:
70
71
72
73

// Register the event handler to be called when keys are pressed
document.onkeydown = function(ev){ keydown(ev, gl, n, viewProjMatrix,
➥u_MvpMatrix, u_NormalMatrix); };
// Draw the robot arm
draw(gl, n, viewProjMatrix, u_MvpMatrix, u_NormalMatrix);

The keydown() function itself is defined at line 80. Before that, at lines 76, 77, and 78, the
definition of global variables used in keydown() is defined:
76 var ANGLE_STEP = 3.0;
// The increments of rotation angle (degrees)
77 var g_arm1Angle = -90.0; // The rotation angle of arm1 (degrees)
78 var g_joint1Angle = 0.0; // The rotation angle of joint1 (degrees)
79
80 function keydown(ev, gl, n, u_MvpMatrix, u_NormalMatrix) {
81 switch (ev.keyCode) {
82 case 38: // Up arrow key -> the positive rotation of joint1 (z-axis)
83
if (g_joint1Angle < 135.0) g_joint1Angle += ANGLE_STEP;
84
break;
...
88 case 39: // Right arrow key -> the positive rotation of arm1 (y-axis)
89
g_arm1Angle = (g_arm1Angle + ANGLE_STEP) % 360;
90
break;
...
95 }
96
// Draw the robot arm
97
draw(gl, n, u_MvpMatrix, u_NormalMatrix);
98 }
ANGLE_STEP at line 76 is used to control how many degrees arm1 and joint1 are rotated
each time the arrow keys are pressed and is set at 3.0 degrees. g_arm1Angle (line 77) and
g_joint1Angle (line 78) are variables that store the current rotation angle of arm1 and

joint1, respectively (see Figure 9.7).

Drawing and Manipulating Objects Composed of Other Objects

331

g_joint1Angle

g_arm1Angle

Figure 9.7

g_joint1Angle and g_arm1Angle

The keydown() function, from line 80, increases or decreases the value of the rotation
angle of arm1 (g_arm1Angle) or joint1 (g_joint1Angle) by ANGLE_STEP, according to which
key is pressed. joint1 can only be rotated through the range from –135 degrees to 135
degrees so that arm2 does not interfere with arm1. Then the whole robot arm is drawn at
line 97 using the function draw().

Draw the Hierarchical Structure (draw())
The draw() function draws the robotic arm according to its hierarchical structure and is
defined at line 177. Two global variables, g_modelMatrix and g_mvpMatrix, are created at
line 175 and will be used in both draw() and drawBox():
174 // Coordinate transformation matrix
175 var g_modelMatrix = new Matrix4(), g_mvpMatrix = new Matrix4();
176
177 function draw(gl, n, viewProjMatrix, u_MvpMatrix, u_NormalMatrix) {
...
181
// Arm1
182
var arm1Length = 10.0; // Length of arm1
183
g_modelMatrix.setTranslate(0.0, -12.0, 0.0);
184
g_modelMatrix.rotate(g_arm1Angle, 0.0, 1.0, 0.0); // Rotate y-axis
185
drawBox(gl, n, viewProjMatrix, u_MvpMatrix, u_NormalMatrix); // Draw
186
187
// Arm2
188
g_modelMatrix.translate(0.0, arm1Length, 0.0); // Move to joint1
189
g_modelMatrix.rotate(g_joint1Angle, 0.0, 0.0, 1.0); // Rotate z-axis
190
g_modelMatrix.scale(1.3, 1.0, 1.3);
// Make it a little thicker
191
drawBox(gl, n, viewProjMatrix, u_MvpMatrix, u_NormalMatrix); // Draw
192 }
332

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As you can see, draw() draws the segments by using drawBox(), starting with the upper
part (arm1) followed by the lower part (arm2).
When drawing each part, the same process is repeated: (1) translation (setTranslate(),
translate()), (2) rotation (rotate()), and (3) drawing the part (drawBox()).
When drawing a hierarchical model performing a rotation, typically you will process from
upper to lower in the order of (1) translation, (2) rotation, and (3) drawing segments.
arm1 is translated to (0.0, –12.0, 0.0) with setTranslate() at line 183 to move to an easily
visible position. Because this arm is rotated around the y-axis, its model matrix (g_modelMatrix) is multiplied by the rotation matrix around the y-axis at line 184. g_arm1Angle
is used here. Once arm1’s coordinate transformation has been completed, you then draw
using the drawBox() function.
Because arm2 is connected to the tip of arm1, as shown in Figure 9.7, it has to be drawn
from the tip of arm1. This can be achieved by translating it along the y-axis in the positive direction by the length of arm1 (arm1Length) and applying the translation to the
model matrix, which is used when drawing arm1 (g_modelMatrix).
This is done as shown in line 188, where the second argument of translate() is
arm1Length. Also notice that the method uses translate() rather than setTranslate()
because arm2 is drawn at the tip of arm1:
187
188
189
190
191

// Arm2
g_modelMatrix.translate(0.0, arm1Length, 0.0); // Move to joint1
g_modelMatrix.rotate(g_joint1Angle, 0.0, 0.0, 1.0); // Rotate z-axis
g_modelMatrix.scale(1.3, 1.0, 1.3); // Make it a little thicker
drawBox(gl, n, viewProjMatrix, u_MvpMatrix, u_NormalMatrix); // Draw

Line 189 handles the rotation of arm2 which, as can be seen, uses g_joint1Angle. You
make arm2 a little thicker at line 190 by scaling it along the x and z direction. This makes
it easier to distinguish between the two arm segments but is not essential to the robotic
arm’s movement.
Now, by updating g_arm1Angle and g_joint1Angle in keydown() as described in the previous section and then invoking draw(), arm1 is rotated by g_arm1Angle and arm2 is, in
addition, rotated by g_joint1Angle.
The drawBox() function is quite simple. It calculates a model view project matrix and
passes it to the u_MvpMatrix variable at lines 199 and 200. Then it just calculates the
normal transformation matrix for shading from the model matrix, sets it to
u_NormalMatrix at lines 203 and 204, and draws the cuboid in Figure 9.6 at line 207.
This basic approach, although used here for only a single joint, can be used for any
complex hierarchical models simply by repeating the process steps used earlier.

Drawing and Manipulating Objects Composed of Other Objects

333

Obviously, our simple robot arm, although modeled on a human arm, is more like a
skeleton than a real arm. A more realistic model of a real arm would require the skin to
be modeled, which is beyond the scope of this book. Please refer to the OpenGL ES 2.0
Programming Guide for more information about skinning.

A Multijoint Model
Here, you will extend JointModel to create MultiJointModel, which draws a multijoint
robot arm consisting of two arm segments, a palm, and two fingers, all of which you can
manipulate using the keyboard. As shown in Figure 9.8, we call the arm extending from
the base arm1, the next segment arm2, and the joint between the two arms joint1. There
is a palm at the tip of arm2. The joint between arm2 and the palm is called joint2. The
two fingers attached at the end of the palm are respectively finger1 and finger2.
finger1 finger2
finger1

joint2

finger2
rotate around the x-axis

arm2

palm
rotate around the y-axis

palm
joint1

arm2
rotate around the z-axis

arm1

arm1
rotate around the y-axis

base

Figure 9.8

base

The hierarchical structure of MultiJointModel

Manipulation of arm1 and joint1 using the arrow keys is the same as JointModel. In addition, you can rotate joint2 (wrist) with the X and Z keys and move (rotate) the two fingers
with the C and V keys. The variables controlling the rotation angle of each part are shown
in Figure 9.9.

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Hierarchical Objects

g_joint3Angle

g_joint2Angle

g_joint1Angle

g_arm1Angle

Figure 9.9

The variables controlling the rotation of segments

Sample Program (MultiJointModel.js)
This program is similar to JointModel, except for extensions to keydown() to handle the
additional control keys, and draw(), which draws the extended hierarchical structure. First
let’s look at keydown() in Listing 9.2.
Listing 9.2

MultiJointModel.js (Code for Key Processing)

1 // MultiJointModel.js
...
76 var ANGLE_STEP = 3.0;
// The increments of rotation angle (degrees)
77 var g_arm1Angle = 90.0;
// The rotation angle of arm1 (degrees)
78 var g_joint1Angle = 45.0; // The rotation angle of joint1 (degrees)
79 var g_joint2Angle = 0.0; // The rotation angle of joint2 (degrees)
80 var g_joint3Angle = 0.0; // The rotation angle of joint3 (degrees)
81
82 function keydown(ev, gl, n, viewProjMatrix, u_MvpMatrix, u_NormalMatrix) {
83
switch (ev.keyCode) {
84
case 40: // Up arrow key -> positive rotation of joint1 (z-axis)
...
95
break;
96
case 90: // Z key -> the positive rotation of joint2
97
g_joint2Angle = (g_joint2Angle + ANGLE_STEP) % 360;
98
break;
99
case 88: // X key -> the negative rotation of joint2
100
g_joint2Angle = (g_joint2Angle - ANGLE_STEP) % 360;
101
break;

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335

102
103
104
105
106

case 86: // V key -> the positive rotation of joint3
if (g_joint3Angle < 60.0) g_joint3Angle = (g_joint3Angle +
➥ANGLE_STEP) % 360;
break;
case 67: // C key -> the negative rotation of joint3
if (g_joint3Angle > -60.0) g_joint3Angle = (g_joint3Angle –
➥ANGLE_STEP) % 360;
break;
default: return;

107
108
109
}
110
// Draw the robot arm
111
draw(gl, n, viewProjMatrix, u_MvpMatrix, u_NormalMatrix);
112 }

keydown() is basically the same as that of JointAngle, but in addition to changing
g_arm1Angle and g_joint1Angle based on key presses, it processes the Z, X, V, and C
keys at lines 96, 99, 102, and 105. These key presses change g_joint2Angle, which is the
rotation angle of joint2, and g_joint3Angle, which is the rotation angle of joint3, respectively. After changing them, it calls draw() at line 111 to draw the hierarchy structure.
Let’s take a look at draw() in Listing 9.3.

Although you are using the same cuboid for the base, arm1, arm2, palm, finger1, and
finger2, the segments are different in width, height, and depth. To make it easy to draw
these segments, let’s extend drawBox() with three more arguments than that used in the
single-joint model:
function drawBox(gl, n, width, height, depth, viewProjMatrix, u_MvpMatrix,
➥u_NormalMatrix)

By specifying the width, height, and depth using the third to fifth argument, this function
draws a cuboid of the specified size with its origin at the center of the bottom surface.
Listing 9.3

MultiJointModel.js (Code for Drawing the Hierarchy Structure)

188 // Coordinate transformation matrix
189 var g_modelMatrix = new Matrix4(), g_mvpMatrix = new Matrix4();
190
191 function draw(gl, n, viewProjMatrix, u_MvpMatrix, u_NormalMatrix) {
192
// Clear color buffer and depth buffer
193
gl.clear(gl.COLOR_BUFFER_BIT | gl.DEPTH_BUFFER_BIT);
194
195
// Draw a base
196
var baseHeight = 2.0;
197
g_modelMatrix.setTranslate(0.0, -12.0, 0.0);
198
drawBox(gl, n, 10.0, baseHeight, 10.0, viewProjMatrix, u_MvpMatrix,
➥u_NormalMatrix);

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199
200
201
202
203
204
295
206
212
213
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242

// Arm1
var arm1Length = 10.0;
g_modelMatrix.translate(0.0, baseHeight, 0.0); // Move onto the base
g_modelMatrix.rotate(g_arm1Angle, 0.0, 1.0, 0.0); // Rotation
drawBox(gl, n, 3.0, arm1Length, 3.0, viewProjMatrix, u_MvpMatrix,
➥u_NormalMatrix); // Draw
// Arm2
...
// A palm
var palmLength = 2.0;
...
// Move to the center of the tip of the palm
g_modelMatrix.translate(0.0, palmLength, 0.0);
// Draw finger1
pushMatrix(g_modelMatrix);
g_modelMatrix.translate(0.0, 0.0, 2.0);
g_modelMatrix.rotate(g_joint3Angle, 1.0, 0.0, 0.0); // Rotation
drawBox(gl, n, 1.0, 2.0, 1.0, viewProjMatrix, u_MvpMatrix, u_NormalMatrix);
g_modelMatrix = popMatrix();
// Draw finger2
g_modelMatrix.translate(0.0, 0.0, -2.0);
g_modelMatrix.rotate(-g_joint3Angle, 1.0, 0.0, 0.0); // Rotation
drawBox(gl, n, 1.0, 2.0, 1.0, viewProjMatrix, u_MvpMatrix, u_NormalMatrix);
}
var g_matrixStack = []; // Array for storing a matrix
function pushMatrix(m) { // Store the specified matrix to the array
var m2 = new Matrix4(m);
g_matrixStack.push(m2);
}
function popMatrix() { // Retrieve the matrix from the array
return g_matrixStack.pop();
}

The draw() function operates in the same way as in JointModel; that is, each part
is handled following the order of (1) translation, (2) rotation, and (3) draw (using
drawBox()). First, because the base is not rotated, after moving to the appropriate position at line 197, it draws a base there with drawBox(). The third to fifth arguments of
drawBox() specify a width of 10, height of 2, and depth of 10, which cause a flat stand to
be drawn.
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337

The arm1, arm2, and palm are each drawn following the same order of (1) translation,
(2) rotation, and (3) draw and by moving down the object hierarchy toward the lower
level in the same manner as JointModel.
The main difference in this sample program is the drawing of finger1 and finger2 from
line 222. Because they do not have a parent-child relationship, a little more care is needed.
In particular, you have to pay attention to the contents of the model matrix. First, let’s
look at finger1, whose position is translated 2.0 along the z-axis direction from the center
of the tip of the palm and rotated around the x-axis. finger1 can be drawn in the order of
(1) translating, (2) rotating, and (3) drawing segments as before. The program is as follows:
g_modelMatrix.translate(0.0, 0.0, 2.0);
g_modelMatrix.rotate(g_joint3Angle, 1.0, 0.0, 0.0); // Rotation
drawBox(gl, n, 1.0, 2.0, 1.0, u_MvpMatrix, u_NormalMatrix);

Next, looking at finger2, if you follow the same procedure a problem occurs. finger2’s
intended position is a translation of –2.0 units along the z-axis direction from the center
of the tip of the palm and rotated around the x-axis. However, because the model matrix
has changed, if you draw finger2, it will be drawn at the tip of finger1.
Clearly, the solution is to restore the model matrix to its state before finger1 was drawn. A
simple way to achieve this is to store the model matrix before drawing finger1 and retrieving it after drawing finger1. This is actually done at lines 222 and 226 and uses the functions pushMatrix() and popMatrix() to store the specified matrix and retrieve it. At line
222, you store the model matrix specified as pushMatrix()’s argument (g_modelMatrix).
Then, after drawing finger1 at lines 223 to 225, you retrieve the old model matrix at line
226, with popMatrix(), and assign it to g_modelMatrix. Now, because the model matrix
has reverted back, you can draw finger2 in the same way as before.
pushMatrix() and popMatrix() are shown next. pushMatrix() stores the matrix specified
as its argument in an array named g_matrixStack at line 234. popMatrix() retrieves the
matrix stored in g_matrixStack and returns it:
234
235
236
237
238
239
240
241
242

var g_matrixStack = []; // Array for storing matrices
function pushMatrix(m) { // Store the specified matrix
var m2 = new Matrix4(m);
g_matrixStack.push(m2);
}
function popMatrix() { // Retrieve a matrix from the array
return g_matrixStack.pop();
}

This approach can be used to draw an arbitrarily long robot arm. It will scale when new
segments are added to the hierarchy. You only need to use pushMatrix() and popMatrix()
when the hierarchy structure is a sibling relation, not a parent-child relation.

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Draw Segments (drawBox())
Finally, let’s take a look at drawBox(), which draws the segments of the robot arm using
the following arguments:
247 function drawBox(gl, n, width, height, depth, viewMatrix, u_MvpMatrix,
➥u_NormalMatrix) {

The third to fifth arguments, width, height, and depth, specify the width, height, and depth
of the cuboid being drawn. As for the remaining argument, viewMatrix is a view matrix,
and u_MvpMatrix and u_NormalMatrix are the arguments for setting the coordinate transformation matrices to the corresponding uniform variables in the vertex shader, just like
JointModel.js. The model view projection matrix is passed to u_MvpMatrix, and the
matrix for transforming the coordinates of the normal, described in the previous section,
is passed to u_NormalMatrix.
The three-dimensional object used here, unlike JointModel, is a cube whose side is 1.0
unit long. Its origin is located at the center of the bottom surface so that you can easily
rotate the arms, the palm, and the fingers. The function drawBox() is shown here:
244 var g_normalMatrix = new Matrix4();// Transformation matrix for normal
245
246 // Draw a cuboid
247 function drawBox(gl, n, width, height, depth, viewProjMatrix,
➥u_MvpMatrix, u_NormalMatrix) {
248
pushMatrix(g_modelMatrix);
// Save the model matrix
249
// Scale a cube and draw
250
g_modelMatrix.scale(width, height, depth);
251
// Calculate model view project matrix and pass it to u_MvpMatrix
252
g_mvpMatrix.set(viewProjMatrix);
253
g_mvpMatrix.multiply(g_modelMatrix);
254
gl.uniformMatrix4fv(u_MvpMatrix, false, g_mvpMatrix.elements);
255
// Calculate transformation matrix for normals and pass it to u_NormalMatrix
...
259
// Draw
260
gl.drawElements(gl.TRIANGLES, n, gl.UNSIGNED_BYTE, 0);
261
g_modelMatrix = popMatrix();
// Retrieve the model matrix
262 }

As you can see, the model matrix is multiplied by a scaling matrix at line 250 so that
the cube will be drawn with the size specified by width, height, and depth. Note that you
store the model matrix at line 248 and retrieve it at line 261 using pushMatrix() and
popMatrix(). Otherwise, when you draw arm2 after arm1, the scaling used for arm1 is left
in the model matrix and affects the drawing of arm2. By retrieving the model matrix at
line 261, which is saved at line 248, the model matrix reverts to the state before scaling
was applied at line 250.

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339

As you can see, the use of pushMatrix() and popMatrix() adds an extra degree of
complexity but allows you to specify only one set of vertex coordinates and use scaling
to create different cuboids. The alternative approach, using multiple objects specified by
different sets of vertices, is also possible. Let’s take a look at how you would program that.

Draw Segments (drawSegment())
In this section, we will explain how to draw segments by switching between buffer objects
in which the vertex coordinates representing the shape of each segment are stored.
Normally, you would need to specify the vertex coordinates, the normal, and the indices
for each segment. However, in this example, because all segments are cuboids, you can
share the normals and indices and simply specify the vertices for each segment. For each
segment (the base, arm1, arm2, palm, and fingers), the vertices are stored in their respective object buffers, which are then switched when drawing the arm parts. Listing 9.4
shows the sample program.
Listing 9.4

MultiJointModel_segment.js

1 // MultiJointModel_segment.js
...
29 function main() {
...
47
var n = initVertexBuffers(gl);
...
57 // Get the storage locations of attribute and uniform variables
58 var a_Position = gl.getAttribLocation(gl.program, 'a_Position');
...
74
draw(gl, n, viewProjMatrix, a_Position, u_MvpMatrix, u_NormalMatrix);
75 }
...
115 var g_baseBuffer = null;
// Buffer object for a base
116 var g_arm1Buffer = null;
// Buffer object for arm1
117 var g_arm2Buffer = null;
// Buffer object for arm2
118 var g_palmBuffer = null;
// Buffer object for a palm
119 var g_fingerBuffer = null;
// Buffer object for fingers
120
121 function initVertexBuffers(gl){
122
// Vertex coordinate (Coordinates of cuboids for all segments)
123
var vertices_base = new Float32Array([ // Base(10x2x10)
124
5.0, 2.0, 5.0, -5.0, 2.0, 5.0, -5.0, 0.0, 5.0, 5.0, 0.0, 5.0,
125
5.0, 2.0, 5.0, 5.0, 0.0, 5.0, 5.0, 0.0,-5.0, 5.0, 2.0,-5.0,
...
129
5.0, 0.0,-5.0, -5.0, 0.0,-5.0, -5.0, 2.0,-5.0, 5.0, 2.0,-5.0
130
]);
131

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132
133
134
138
139
159
166
167
168
169

var vertices_arm1 = new Float32Array([ // Arm1(3x10x3)
1.5, 10.0, 1.5, -1.5, 10.0, 1.5, -1.5, 0.0, 1.5, 1.5, 0.0, 1.5,
1.5, 10.0, 1.5, 1.5, 0.0, 1.5, 1.5, 0.0,-1.5, 1.5, 10.0,-1.5,
...
1.5, 0.0,-1.5, -1.5, 0.0,-1.5, -1.5, 10.0,-1.5, 1.5, 10.0,-1.5
]);
...
var vertices_finger = new Float32Array([ // Fingers(1x2x1)
...
]);

// normals
var normals = new Float32Array([
...
176
]);
177
178
// Indices of vertices
179
var indices = new Uint8Array([
180
0, 1, 2,
0, 2, 3,
// front
181
4, 5, 6,
4, 6, 7,
// right
...
185
20,21,22, 20,22,23
// back
186
]);
187
188
// Write coords to buffers, but don't assign to attribute variables
189
g_baseBuffer = initArrayBufferForLaterUse(gl, vertices_base, 3, gl.FLOAT);
190
g_arm1Buffer = initArrayBufferForLaterUse(gl, vertices_arm1, 3, gl.FLOAT);
...
193
g_fingerBuffer = initArrayBufferForLaterUse(gl, vertices_finger, 3, gl.FLOAT);
...
196
// Write normals to a buffer, assign it to a_Normal, and enable it
197
if (!initArrayBuffer(gl, 'a_Normal', normals, 3, gl.FLOAT)) return null;
198
199
// Write indices to a buffer
200
var indexBuffer = gl.createBuffer();
...
205
gl.bindBuffer(gl.ELEMENT_ARRAY_BUFFER, indexBuffer);
206
gl.bufferData(gl.ELEMENT_ARRAY_BUFFER, indices, gl.STATIC_DRAW);
207
208
return indices.length;
209 }
...
255 function draw(gl, n, viewProjMatrix, a_Position, u_MvpMatrix, u_NormalMatrix) {
...

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341

259
260
261
262
263
264
265
266
267
268

// Draw a base
var baseHeight = 2.0;
g_modelMatrix.setTranslate(0.0, -12.0, 0.0);
drawSegment(gl, n, g_baseBuffer, viewProjMatrix, a_Position,
➥u_MvpMatrix, u_NormalMatrix);
// Arm1
var arm1Length = 10.0;
g_modelMatrix.translate(0.0, baseHeight, 0.0); // Move to the tip of the base
g_modelMatrix.rotate(g_arm1Angle, 0.0, 1.0, 0.0); // Rotate y-axis
drawSegment(gl, n, g_arm1Buffer, viewProjMatrix, a_Position,
➥u_MvpMatrix, u_NormalMatrix);

269
270

// Arm2
...
292
// Finger2
...
295
drawSegment(gl, n, g_fingerBuffer, viewProjMatrix, a_Position,
➥u_MvpMatrix, u_NormalMatrix);
296 }
...
310 // Draw segments
311 function drawSegment(gl, n, buffer, viewProjMatrix, a_Position,
➥u_MvpMatrix, u_NormalMatrix) {
312
gl.bindBuffer(gl.ARRAY_BUFFER, buffer);
313 // Assign the buffer object to the attribute variable
314 gl.vertexAttribPointer(a_Position, buffer.num, buffer.type, false, 0, 0);
315
// Enable the assignment
316
gl.enableVertexAttribArray(a_Position);
317
318
// Calculate the model view project matrix and set it to u_MvpMatrix
...
322
// Calculate matrix for normal and pass it to u_NormalMatrix
...
327
gl.drawElements(gl.TRIANGLES, n, gl.UNSIGNED_BYTE, 0);
328 }

The key points in this program are (1) creating the separate buffer objects that contain the
vertex coordinates for each segment, (2) before drawing each segment, assigning the corresponding buffer object to the attribute variable a_Position, and (3) enabling the buffer
and then drawing the segment.
The main() function from line 29 in the JavaScript code follows the same steps as before.
Switching between buffers for the different segments is added to initVertexBuffers(),

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called at line 47. The stored location of a_Position is retrieved at line 58, and then draw()
is called at line 73.
Let’s examine initVertex(), defined at line 121. Lines 115 to 119 declare the buffer
objects as global variables, used to store the vertex coordinates of each segment. Within
the function, one of the main differences from MultiJointModel.js is the definition of
the vertex coordinates from line 123. Because you are not using a single cuboid transformed differently for the different segments, you need to define the vertex coordinates
for all the parts separately (for example, the base (vertices_base) at line 123, coordinates
for arm1 (vertices_arm1), at line 132. The actual creation of the buffer objects for each
part occurs in the function initArrayBufferForLaterUse() from line 189 to 193. This
function is shown here:
211 function initArrayBufferForLaterUse(gl, data, num, type){
212
var buffer = gl.createBuffer();
// Create a buffer object
...
217
// Write data to the buffer object
218
gl.bindBuffer(gl.ARRAY_BUFFER, buffer);
219
gl.bufferData(gl.ARRAY_BUFFER, data, gl.STATIC_DRAW);
220
221
// Store information to assign it to attribute variable later
222
buffer.num = num;
223
buffer.type = type;
224
225
return buffer;
226 }
initArrayBufferForLaterUse() simply creates a buffer object at line 212 and writes data
to it at lines 218 and 219. Notice that assigning it to an attribute variable (gl.vertex
AttribPointer()) and enabling the assignment (gl.enableVertexAttribAray()) are not

done within the function but later, just before drawing. To assign the buffer object to the
attribute variable a_Position later, the data needed is stored as properties of the buffer
object at lines 222 and 223.
Here you take advantage of an interesting feature of JavaScript that allows you to freely
add new properties of an object and assign data to them. You can do this by simply
appending the .property-name to the object name and assigning a value. Using this
feature, you store the number of items in the num property (line 222), and the type in the
type property (line 223). Of course, you can access the contents of the newly made properties using the same name. Note, you must be careful when referring to properties created
in this way, because JavaScript gives no error indications even if you misspell only one
character in the property name. Equally, be aware that, although convenient, appending
properties has a performance overhead. A better approach, user-defined types, is explained
in Chapter 10, “Advanced Techniques,” but let’s stick with this approach for now.

Drawing and Manipulating Objects Composed of Other Objects

343

Finally, the draw() function, invoked at line 255, is the same as used in MultiJointModel
in terms of drawing parts according to the hierarchical structure, but it’s different in
terms of using drawSegment() to draw each segment. In particular, the third argument of
drawSegment(), shown next, is the buffer object in which the vertex coordinates of the
parts are stored.
262

drawSegment(gl, n, g_baseBuffer, viewProjMatrix, u_MvpMatrix, u_NormalMatrix);

This function is defined at line 311 and operates as follows. It assigns a buffer object to
the attribute variable a_Position and enables it at lines 312 to 316 before drawing at line
327. Here, num and type, which are just stored as buffer object properties, are used.
310
311
312
313
314
315
316
317
318

// Draw segments
function drawSegment(gl, n, buffer, viewProjMatrix, a_Position,
➥u_MvpMatrix, u_NormalMatrix) {
gl.bindBuffer(gl.ARRAY_BUFFER, buffer);
// Assign the buffer object to the attribute variable
gl.vertexAttribPointer(a_Position, buffer.num, buffer.type, false, 0, 0);
// Enable the assignment
gl.enableVertexAttribArray(a_Position);
// Calculate model view project matrix and set it to u_MvpMatrix
...

322

// Calculate transformation matrix for normal and set it to u_NormalMatrix
...

327
328

gl.drawElements(gl.TRIANGLES, n, gl.UNSIGNED_BYTE, 0);
}

This time you don’t need to scale objects with the model matrix because you have
prepared the vertex coordinates per part, so there is no need to store and retrieve the
matrix. Therefore, pushMatrix() and popMatrix() are not necessary.

Shader and Program Objects: The Role of
initShaders()
Finally, before we wrap up this chapter, let’s examine one of the convenience functions
defined for this book: initShaders(). This function has been used in all the sample
programs and has hidden quite a lot of complex detail about setting up and using shaders.
We have deliberately left this explanation to the end of this chapter to ensure you have a
good understanding of the basics of WebGL before tackling some of these complex details.
We should note that it’s not actually necessary to master these details. For some readers
it will be sufficient to simply reuse the initShaders() function we supply and skip this
section. However, for those who are interested, let’s take a look.

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initShaders() carries out the routine work to make shaders available in WebGL. It

consists of seven steps:
1. Create shader objects (gl.createShader()).
2. Store the shader programs (to avoid confusion, we refer to them as “source code”) in
the shader objects (g.shaderSource()).
3. Compile the shader objects (gl.compileShader()).
4. Create a program object (gl.createProgram()).
5. Attach the shader objects to the program object (gl.attachShader()).
6. Link the program object (gl.linkProgram()).
7. Tell the WebGL system the program object to be used (gl.useProgram()).
Each step is simple but when combined can appear complex, so let’s take a look at them
one by one. First, as you know from earlier, two types of objects are necessary to use
shaders: shader objects and program objects.
Shader object

A shader object manages a vertex shader or a fragment shader. One
shader object is created per shader.

Program object

A program object is a container that manages the shader objects. A vertex
shader object and a fragment shader object (two shader objects in total)
must be attached to a program object in WebGL.

The relationship between a program object and shader objects is shown in Figure 9.10.

A program object

shader object
(vertex shader)

Figure 9.10

shader object
(fragment shader)

The relationship between a program object and shader objects

Using this information, let’s discuss the preceding seven steps sequentially.

Create Shader Objects (gl.createShader())
All shader objects have to be created with a call to gl.createShader() before using them.

Shader and Program Objects: The Role of initShaders()

345

gl.createShader(type)
Create a shader of the specified type.
Parameters

type

Specifies the type of shader object to be created: either
gl.VERTEX_SHADER (a vertex shader) or gl.FRAGMENT_
SHADER (a fragment shader).

Return value

Non-null

The created shader object.

null

The creation of the shader object failed.

INVALID_ENUM

The specified type is none of the above.

Errors

gl.createShader() creates a vertex shader or a fragment shader according to the specified
type. If you do not need the shader any more, you can delete it with gl.deleteShader().

gl.deleteShader(shader)
Delete the shader object.
Parameters

shader

Return value

None

Errors

None

Specifies the shader object to be deleted.

Note that the specified shader object is not deleted immediately if it is still in use (that is,
it is attached to a program object using gl.attachShader(), which is discussed in a few
pages). The shader object specified as an argument of gl.deleteShader() will be deleted
when a program object no longer uses it.

Store the Shader Source Code in the Shader Objects
(g.shaderSource())
A shader object has storage to store the shader source code (written as a string in the
JavaScript program or in the separate file; see Appendix F, “Loading Shader Programs from
Files”). You use gl.shaderSource() to store the source code in a shader object.

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gl.shaderSource(shader, source)
Store the source code specified by source in the shader object specified by shader. If any
source code was previously stored in the shader object, it is replaced by new source code.
Parameters

shader

Specifies the shader object in which the program is stored.

source

Specifies the shader source code (string)

Return value

None

Errors

None

Compile Shader Objects (gl.compileShader())
After storing the shader source code in the shader object, you have to compile it so that
it can be used in the WebGL system. Unlike JavaScript, and like C or C++, shaders need
to be compiled before use. In this process, the source code stored in a shader object is
compiled to executable format (binary) and kept in the WebGL system. Use gl.compileShader() to compile. Note, if you replace the source code in the shader object with a call
to gl.shaderSource() after compiling, the compiled binary kept in the shader object is
not replaced. You have to recompile it explicitly.

gl.compileShader(shader)
Compile the source code stored in the shader object specified by shader.
Parameters

shader

Return Value

None

Errors

None

Specifies the shader object in which the source code to be
compiled is stored.

When executing gl.compileShader(), it is possible a compilation error occurs due to
mistakes in the source code. You can check for such errors, as well as the status of the
shader object, using gl.getShaderParameter().

Shader and Program Objects: The Role of initShaders()

347

gl.getShaderParameter(shader, pname)
Get the information specified by pname from the shader object specified by shader.
Parameters

shader

Specifies the shader object.

pname

Specifies the information to get from the shader:
gl.SHADER_TYPE, gl.DELETE_STATUS, or
gl.COMPILE_STATUS.

Return value

The following depending on pname:
gl.SHADER_TYPE

The type of shader (gl.VERTEX_SHADER or gl.FRAGMENT_
SHADER)

Errors

gl.DELETE_
STATUS

Whether the deletion has succeeded (true or false)

gl.COMPILE_
STATUS

Whether the compilation has succeeded (true or false)

INVALID_ENUM

pname is none of the above values.

To check whether the compilation succeeded, you can call gl.getShaderParameter() with
gl.COMPILE_STATUS specified in pname.

If the compilation has failed, gl.getShaderParameter() returns false, and the error information is written in the information log for the shader in the WebGL system. This information can be retrieved with gl.getShaderInfoLog().

gl.getShaderInfoLog(shader)
Retrieve the information log from the shader object specified by shader.
Parameters

shader

Specifies the shader object from which the information log is
retrieved.

Return value

non-null

The string containing the logged information.

null

Any errors are generated.

Errors

None

Although the exact details of the logged information is implementation specific, almost
all WebGL systems return error messages containing the line numbers where the compiler

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has detected the errors in the program. For example, assume that you compiled a fragment
shader program as follows:
var FSHADER_SOURCE =
'void main() {\n' +
' gl.FragColor = vec4(1.0, 0.0, 0.0, 1.0);\n' +
'}\n';

Because the second line is incorrect in this case (gl. must be gl_), the error messages
displayed in the JavaScript console of Chrome will be similar to those shown in Figure 9.11.

Figure 9.11

A compile error in a shader

The first message indicates that gl at line 2 is undeclared.
failed to compile shader: ERROR: 0:2: 'gl' : undeclared identifier
cuon-utils.js:88

The reference to cuon-utils.js:88 on the right means that the error has been detected in
gl.getShaderInfoLog(), which was invoked at line 88 of the cuon-utils.js file, where
initShaders() is defined.

Create a Program Object (gl.createProgram())
As mentioned before, a program object is a container to store the shader objects and is
created by gl.createProgram(). You are already familiar with this program object because
it is the object you pass as the first argument of gl.getAttribLocation() and
gl.getUniformLocation().

gl.createProgram()
Create a program object.
Parameters

None

Return value

non-null

The newly created program object.

null

Failed to create a program object.

Errors

None

Shader and Program Objects: The Role of initShaders()

349

A program object can be deleted by using gl.deleteProgram().

gl.deleteProgram(program)
Delete the program object specified by program. If the program object is not referred to
from anywhere, it is deleted immediately. Otherwise, it will be deleted when it is no
longer referred to.
Parameters

program Specifies the program object to be deleted.

Return value

None

Errors

None

Once the program object has been created, you attach the two shader objects to it.

Attach the Shader Objects to the Program Object (gl.attachShader())
Because you always need two shaders in WebGL—a vertex shader and a fragment shader—
you must attach both of them to the program object with gl.attachShader().

gl.attachShader(program, shader)
Attach the shader object specified by shader to the program object specified by program.
Parameters

program

Specifies the program object.

shader

Specifies the shader object to be attached to
program.

Return value

None

Errors

INVALID_OPERATION

Shader had already been attached to program.

It is not necessary to compile or store any source code before it is attached to the program
object. You can detach the shader object with gl.detachShader().

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gl.detachShader(program, shader)
Detach the shader object specified by shader from the program object specified by
program.
Parameters

program

Specifies the program object.

shader

Specifies the shader object to be detached from
program.

Return value

None

Errors

INVALID_OPERATION

shader is not attached to program.

Link the Program Object (gl.linkProgram())
After attaching shader objects to a program object, you need to link the shader objects.
You use gl.linkProgram() to link the shader objects in the program object.

gl.linkProgram(program)
Link the program object specified by program.
Parameters

program

Return value

None

Errors

None

Specifies the program object to be linked.

During linking, various constraints of the WebGL system are checked: (1) when varying
variables are declared in a vertex shader, whether varying variables with the same names
and types are declared in a fragment shader, (2) whether a vertex shader has written data
to varying variables used in a fragment shader, (3) when the same uniform variables
are used in both a vertex shader and a fragment shader, whether their types and names
match, (4) whether the numbers of attribute variables, uniform variables, and varying
variables does not exceed an upper limit, and so on.
After linking the program object, it is always good programming practice to check whether
it succeeded. The result of linking can be confirmed with gl.getProgramParameters().

Shader and Program Objects: The Role of initShaders()

351

gl.getProgramParameter(program, pname)
Return information about pname for the program object specified by program. The return
value differs depending on pname.
Parameters

Return value

Errors

program

Specifies the program object.

pname

Specifies any one of gl.DELETE_STATUS, gl.LINK_STATUS,
gl.VALIDATE_STATUS, gl.ATTACHED_SHADERS, gl.ACTIVE_
ATTRIBUTES, or gl.ACTIVE_UNIFORMS.

Depending on pname, the following values can be returned:
gl.DELETE_STATUS

Whether the program has been
deleted (true or false)

gl.LINK_STATUS

Whether the program was linked
successfully (true or false)

gl.VALIDATE_STATUS

Whether the program was validated
successfully (true or false)1

gl.ATTACHED_SHADERS

The number of attached shader
objects

gl.ACTIVE_ATTRIBUTES

The number of attribute variables in
the vertex shader

gl.ACTIVE_UNIFORMS

The number of uniform variables

INVALID_ENUM

pname is none of the above values.

If linking succeeded, you are returned an executable program object. Otherwise, you can
get the information about the linking from the information log of the program object
with gl.getProgramInfoLog().

gl.getProgramInfoLog(program)
Retrieve the information log from the program object specified by program.
Parameters

program

Specifies the program object from which the information log is
retrieved.

Return value

The string containing the logged information

Errors

None

A program object may fail to execute even if it was linked successfully, such as if no texture units are
set for the sampler. This can only be detected when drawing, not when linking. Because this check
takes time, check for these errors only when debugging and turn off otherwise.
1

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Tell the WebGL System Which Program Object to Use
(gl.useProgram())
The last step is to tell the WebGL system which program object to use when drawing by
making a call to gl.useProgram().

gl.useProgram(program)
Tell the WebGL system that the program object specified by program will be used.
Parameters

program Specifies the program object to be used.

Return value

None

Errors

None

One powerful feature of this function is that you can use it during drawing to switch
between multiple shaders prepared in advance. This will be discussed and used in
Chapter 10.
With this final step, the preparation for drawing with the shaders is finished. As you have
seen, initShaders() hides quite a lot of detail and can be safely used without worrying
about this detail. Essentially, once executed, the vertex and fragment shaders are set up
and can be used with calls to gl.drawArrays() or gl.drawElements().
Now that you have an understanding of the steps and appropriate WebGL functions used
in initShaders(), let’s take a look at the program flow of initShaders() as defined in
cuon-utils.js.

The Program Flow of initShaders()
initShaders() is composed of two main functions: createProgram(), which creates a
linked program object, and loadShader(), called from createProgram(), which creates the
compiled shader objects. Both are defined in cuon-utils.js. Here, you will work through
initShader() in order from the top (see Listing 9.5). Note that in contrast to the normal
code samples used in the book, the comments in this code are in the JavaDoc form, which
is used in the convenience libraries.

Listing 9.5

initShaders()

1 // cuon-utils.js
2 /**
3 * Create a program object and make current
4 * @param gl GL context
5 * @param vshader a vertex shader program (string)
6 * @param fshader a fragment shader program (string)
7 * @return true, if the program object was created and successfully made current

Shader and Program Objects: The Role of initShaders()

353

8 */
9 function initShaders(gl, vshader, fshader) {
10
var program = createProgram(gl, vshader, fshader);
...
16
gl.useProgram(program);
17
gl.program = program;
18
19
return true;
20 }

First, initShaders() creates a linked program object with createProgram() at line 10 and
tells the WebGL system to use the program object at line 16. Then it sets the program
object to the property named program of the gl object.
Next, look at createProgram() in Listing 9.6.
Listing 9.6

createProgram()

22 /**
23 * Create the linked program object
24 * @param gl GL context
25 * @param vshader a vertex shader program(string)
26 * @param fshader a fragment shader program(string)
27 * @return created program object, or null if the creation has failed.
28 */
29 function createProgram(gl, vshader, fshader) {
30
// Create shader objects
31
var vertexShader = loadShader(gl, gl.VERTEX_SHADER, vshader);
32
var fragmentShader = loadShader(gl, gl.FRAGMENT_SHADER, fshader);
...
37
// Create a program object
38
var program = gl.createProgram();
...
43
// Attach the shader objects
44
gl.attachShader(program, vertexShader);
45
gl.attachShader(program, fragmentShader);
46
47
// Link the program object
48
gl.linkProgram(program);
49
50
// Check the result of linking
51
var linked = gl.getProgramParameter(program, gl.LINK_STATUS);
...
60
return program;
61 }

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The function createProgram() creates the shader objects for the vertex and the fragment shaders, which are loaded using loadShader() at lines 31 and 32. The shader
object returned from loadShader() contains the stored shader source code and compiled
versions.
The program object, to which the shader objects created here will be attached, is created
at line 38, and the vertex and fragment shader objects are attached at lines 44 and 45.
Then createProgram() links the program object at line 48 and checks the result at line 51.
If the linking has succeeded, it returns the program object at line 60.
Finally, let’s look at loadShader()(Listing 9.7) which was invoked at lines 31 and 32 from
within createProgram().
Listing 9.7

loadShader()

63 /**
64 * Create a shader object
65 * @param gl GL context
66 * @param type the type of the shader object to be created
67 * @param source a source code of a shader (string)
68 * @return created shader object, or null if the creation has failed.
69 */
70 function loadShader(gl, type, source) {
71
// Create a shader object
72
var shader = gl.createShader(type);
...
78
// Set source codes of the shader
79
gl.shaderSource(shader, source);
80
81
// Compile the shader
82
gl.compileShader(shader);
83
84
// Check the result of compilation
85
var compiled = gl.getShaderParameter(shader, gl.COMPILE_STATUS);
...
93
return shader;
94 }

First loadShader() creates a shader object at line 72. It associates the source code to the
object at line 79 and compiles it at line 82. Finally, it checks the result of compilation
at line 85 and, if no errors have occurred, returns the shader object to createShader(),
which attaches it to the program object.

Shader and Program Objects: The Role of initShaders()

355

Summary
This chapter is the final one to explore basic features of WebGL. It looked at how to draw
and manipulate complex 3D objects composed of multiple segments organized in a hierarchical structure. This technique is important for understanding how to use simple 3D
objects like cubes or blocks to build up more complex objects like robots or game characters. In addition, you looked at one of the most complex convenience functions we have
provided for this book, initShaders(), which has been treated as a black box up until
now. You saw the details of how shader objects are created and managed by program
objects, so you have a better sense of the internal structure of shaders and how WebGL
manages them through program objects.
At this stage you have a full understanding of WebGL and are capable of writing your
own complex 3D scenes using the expressive power of WebGL. In the next chapter, we
will outline various advanced techniques used in 3D graphics and leverage what you have
learned so far to show how WebGL can support these techniques.

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Chapter 10

Advanced Techniques

This chapter includes a “grab-bag” of interesting techniques that you should find useful
for creating your WebGL applications. The techniques are mostly stand-alone, so you
can select and read any section that interests you. Where there are dependencies, they
are clearly identified. The explanations in this chapter are terse in order to include as
many techniques as possible. However, the sample programs on the website include
comprehensive comments, so please refer to them as well.

Rotate an Object with the Mouse
When creating WebGL applications, sometimes you want users to be able to control 3D
objects with the mouse. In this section, you construct a sample program RotateObject,
which allows users to rotate a cube by dragging it with the mouse. To make the
program simple, it uses a cube, but the basic method is applicable to any object. Figure
10.1 shows a screen shot of the cube that has a texture image mapped onto it.

Figure 10.1

A screen shot of RotateObject

How to Implement Object Rotation
Rotating a 3D object is simply the application of a technique you’ve already studied for
2D objects—transforming the vertex coordinates by using the model view projection
matrix. The process requires you to create a rotation matrix based on the mouse movement, change the model view projection matrix, and then transform the coordinates by
using the matrix.
You can obtain the amount of mouse movement by simply recording the position where
the mouse is initially clicked and then subtracting that position from the new position as
the mouse moves. Clearly, an event handler will be needed to calculate the mouse movement, and then this will be converted into an angle that will rotate the object. Let’s take a
look at the sample program.

Sample Program (RotateObject.js)
Listing 10.1 shows the sample program. As you can see, the shaders do not do anything
special. Line 9 in the vertex shader transforms the vertex coordinates by using the model
view projection matrix, and line 10 maps the texture image onto the cube.
Listing 10.1
1
2
3
8
9
10
11
24
42
61
62
63
64
65
66
74
75
76
77

358

RotateObject.js

// RotateObject.js
// Vertex shader program
var VSHADER_SOURCE =
...
'void main() {\n' +
' gl_Position = u_MvpMatrix * a_Position;\n' +
' v_TexCoord = a_TexCoord;\n' +
'}\n';
...
function main() {
...
var n = initVertexBuffers(gl);
...
viewProjMatrix.setPerspective(30.0, canvas.width / canvas.height,
➥1.0, 100.0);
viewProjMatrix.lookAt(3.0, 3.0, 7.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0);
// Register the event handler
var currentAngle = [0.0, 0.0]; // [x-axis, y-axis] degrees
initEventHandlers(canvas, currentAngle);
...
var tick = function() {
// Start drawing
draw(gl, n, viewProjMatrix, u_MvpMatrix, currentAngle);
requestAnimationFrame(tick, canvas);
};

CHAPTER 10

Advanced Techniques

78
79
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178

tick();
}
...
function initEventHandlers(canvas, currentAngle) {
var dragging = false;
// Dragging or not
var lastX = -1, lastY = -1;
// Last position of the mouse
canvas.onmousedown = function(ev) { // Mouse is pressed
var x = ev.clientX, y = ev.clientY;
// Start dragging if a mouse is in 
var rect = ev.target.getBoundingClientRect();
if (rect.left <= x && x < rect.right && rect.top <= y && y < rect.bottom) {
lastX = x; lastY = y;
dragging = true;
}
};
// Mouse is released
canvas.onmouseup = function(ev) { dragging = false; };
canvas.onmousemove = function(ev) {
// Mouse is moved
var x = ev.clientX, y = ev.clientY;
if (dragging) {
var factor = 100/canvas.height; // The rotation ratio
var dx = factor * (x - lastX);
var dy = factor * (y - lastY);
// Limit x-axis rotation angle to -90 to 90 degrees
currentAngle[0] = Math.max(Math.min(currentAngle[0] + dy, 90.0), -90.0);
currentAngle[1] = currentAngle[1] + dx;
}
lastX = x, lastY = y;
};
}
var g_MvpMatrix = new Matrix4(); // The model view projection matrix
function draw(gl, n, viewProjMatrix, u_MvpMatrix, currentAngle) {
// Calculate the model view projection matrix
g_MvpMatrix.set(viewProjMatrix);
g_MvpMatrix.rotate(currentAngle[0], 1.0, 0.0, 0.0); // x-axis
g_MvpMatrix.rotate(currentAngle[1], 0.0, 1.0, 0.0); // y-axis
gl.uniformMatrix4fv(u_MvpMatrix, false, g_MvpMatrix.elements);
gl.clear(gl.COLOR_BUFFER_BIT | gl.DEPTH_BUFFER_BIT);
gl.drawElements(gl.TRIANGLES, n, gl.UNSIGNED_BYTE, 0);
}

Rotate an Object with the Mouse

359

At lines 61 and 62 of main() in JavaScript, the view projection matrix is calculated in
advance. You will have to change the model matrix on-the-fly according to the amount of
mouse movement.
The code from line 65 registers the event handlers, a key part of this sample program. The
variable currentAngle is initialized at line 65 and used to hold the current rotation angle.
Here, it is an array because it needs to handle two rotation angles around the x-axis and
y-axis. The actual registration of the event handlers is done inside initEventHandlers(),
called at line 66. It draws the cube using the function tick() that is defined from line 74.
initEventHandlers() is defined at line 138. The code from line 142 handles mouse down,

the code from line 152 handles mouse up, and the code from line 154 handles the mouse
movement.
The processing when the mouse button is first pushed at line 142 is simple. Line 146
checks whether the mouse has been pressed inside the  element. If it is inside the
, line 147 saves that position in lastX and lastY. Then the variable dragging,
which indicates dragging has begun, is set to true at line 148.
The processing of the mouse button release at line 152 is simple. Because this indicates
that dragging is done, the code simply sets the variable dragging back to false.
The processing from line 154 is the critical part and tracks the movement of the mouse.
Line 156 checks whether dragging is taking place and, if it is, lines 158 and 159 calculate
how long it has moved, storing the results to dx and dy. These values are scaled, using
factor, which is a function of the canvas size. Once the distance dragged has been calculated, it can be used to determine the new angle by directly adding to the current angles
at line 161 and 162. The code limits rotation from –90 to +90 degrees simply to show the
technique; you are free to remove this. Because the mouse has moved, its position is saved
in lastX and lastY.
Once you have successfully transformed the movement of the mouse into a rotation
angle, you can let the rotation matrix handle the updates and draw the results using
tick(). These operations are done at lines 172 and 173.
This quick review of a technique to calculate the rotation angle is only one approach.
Others, such as placing virtual track balls around the object, are described in detail in the
book 3D User Interfaces.

Select an Object
When your application requires users to be able to control 3D objects interactively, you
will need a technique to allow users to select objects. There are many uses of this technique, such as selecting a 3D button created by a 3D model instead of the conventional
2D GUI button, or selecting a photo among multiple photos in a 3D scene.

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Selecting a 3D object is generally more complex than selecting a 2D one because of the
mathematics required to determine if the mouse is over a nonregular shape. However,
you can use a simple trick, shown in the sample program, to avoid that complexity. In
this sample, PickObject, the user can click a rotating cube, which causes a message to be
displayed (see Figure 10.2). First, run the sample program and experiment with it for a
while to get the feeling of how it works.

Figure 10.2

PickObject

Figure 10.2 shows with the message displayed when clicking the cube. The message says,
“The cube was selected!” Also check what happens when you click the black part of the
background.

How to Implement Object Selection
This program goes through the following steps to check whether the cube was clicked:
1. When the mouse is pressed, draw the cube with a single color “red” (see the middle
of Figure 10.3).
2. Read the pixel value (color) of the selected point.
3. Redraw the cube with its original color (right in Figure 10.3).
4. If the color of the pixel is red, display, “The cube was selected!”
When the cube is drawn with a single color (red in this case), you can quickly see which
part of the drawing area the cube occupies. After reading the pixel value at the position
of the mouse pointer when the mouse is clicked, you can determine that the mouse was
above the cube if the pixel color is red.

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361

Click

Figure 10.3

The object drawn at the point of mouse pressing

To ensure that the viewer doesn’t see the cube flash red, you need to draw and redraw in
the same function. Let’s take a look at the actual sample program.

Sample Program (PickObject.js)
Listing 10.2 shows the sample program. The processing in this sample mainly takes place
in the vertex shader. To implement step 1, you must inform the vertex shader that the
mouse has been clicked so that it draws the cube red. The variable u_Clicked transmits
this information and declared at line 7 in the vertex shader. When the mouse is pressed,
u_Clicked is set to true in the JavaScript and tested at line 11. If true, the color red is
assigned to v_Color; if not, the color of the cube (a_Color) is directly assigned to v_Color.
This turns the cube red when the mouse is pressed.
Listing 10.2
1
2
3
6
7
8
9
10
11
12
13
14
15
16
17
18

362

PickObject.js

// PickObject.js
// Vertex shader program
var VSHADER_SOURCE =
...
'uniform mat4 u_MvpMatrix;\n' +
'uniform bool u_Clicked;\n' + // Mouse is pressed
'varying vec4 v_Color;\n' +
'void main() {\n' +
' gl_Position = u_MvpMatrix * a_Position;\n' +
' if (u_Clicked) {\n' + // Draw in red if mouse is pressed
'
v_Color = vec4(1.0, 0.0, 0.0, 1.0);\n' +
' } else {\n' +
'
v_Color = a_Color;\n' +
' }\n' +
'}\n';
// Fragment shader program

CHAPTER 10

Advanced Techniques

<-(1)

25
30
60
71
72
73
74
75
76
77
78
79
80
81

...
' gl_FragColor = v_Color;\n' +
...
function main() {
...
var u_Clicked = gl.getUniformLocation(gl.program, 'u_Clicked');
...
gl.uniform1i(u_Clicked, 0); // Pass false to u_Clicked
var currentAngle = 0.0; // Current rotation angle
// Register the event handler
canvas.onmousedown = function(ev) {
// Mouse is pressed
var x = ev.clientX, y = ev.clientY;
var rect = ev.target.getBoundingClientRect();
if (rect.left <= x && x < rect.right && rect.top <= y && y < rect.bottom) {
// Check if it is on object
var x_in_canvas = x - rect.left, y_in_canvas = rect.bottom - y;
var picked = check(gl, n, x_in_canvas, y_in_canvas, currentAngle,
➥u_Clicked, viewProjMatrix, u_MvpMatrix);
if (picked) alert('The cube was selected! ');
<-(4)
}
}
...

82
83
84
92
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162

}
...
function check(gl, n, x, y, currentAngle, u_Clicked, viewProjMatrix,
➥u_MvpMatrix) {
var picked = false;
gl.uniform1i(u_Clicked, 1); // Draw the cube with red
draw(gl, n, currentAngle, viewProjMatrix, u_MvpMatrix);
// Read pixel at the clicked position
var pixels = new Uint8Array(4); // Array for storing the pixels
gl.readPixels(x, y, 1, 1, gl.RGBA, gl.UNSIGNED_BYTE, pixels);
<-(2)
if (pixels[0] == 255)
picked = true;

// The mouse in on cube if pixels[0] is 255

gl.uniform1i(u_Clicked, 0); // Pass false to u_Clicked: redraw cube
draw(gl, n, currentAngle, viewProjMatrix, u_MvpMatrix); //

<-(3)

return picked;
}

Let’s take a look from line 30 of main() in JavaScript. Line 60 obtains the storage location
for u_Clicked, and line 71 assigns the initial value of u_Clicked to be false.
Select an Object

363

Line 75 registers the event handler to be called when the mouse has been clicked. This
event handler function does a sanity check to see if the clicked position is inside the
 element at line 78. Then it calls to check() at line 81 if it is. This function checks
whether the position, specified by the third and fourth arguments, is on the cube (see next
paragraph). If so, it returns true which causes a message to be displayed at line 82.
The function check() begins from line 147. This function processes steps (2) and (3) from
the previous section together. Line 149 informs the vertex shader that the click event has
occurred by passing 1 (true) to u_Clicked. Then line 150 draws the cube with the current
rotation angle. Because u_Clicked is true, the cube is drawn in red. Then the pixel value
of the clicked position is read from the color buffer at line 153. The following shows the
gl.readPixels() function used here.

gl.readPixels(x, y, width, height, format, type, pixels)
Read a block of pixels from the color buffer1 and store it to the array pixels. x, y, width,
and height define the block as a rectangle.
Parameters

x, y

Specify the position of the first pixel that is read from the
buffer.

width, height Specify the dimensions of the pixel rectangle.
format

Specifies the format of the pixel data. gl.RGBA must be
specified.

type

Specifies the data type of the pixel data. gl.UNSIGNED_BYTE
must be specified.

pixels

Specifies the typed array (Uint8Array) for storing the pixel
data.

Return value

None

Errors

INVALID_VALUE: pixels is null. Either width or height is negative.
INVALID_OPERATION: pixels is not large enough to store the pixel data.
INVALID_ENUM: format or type is none of the above values.

The pixel value that was read is stored in the pixels array. This array is defined at line
152, and the R, G, B, and A values are stored in pixels[0], pixels[1], pixels[2], and
pixels[3], respectively. Because, in this sample program, you know that the only colors
used are red for the cube and black for the background, you can see if the mouse is on
the cube by checking the values for pixels[0]. This is done at line 155, and if it is red, it
changes picked to true.
1

If a framebuffer object is bound to gl.FRAMEBUFFER, this method reads the pixel values from the
object. We explain the object in the later section “Use What You’ve Drawn as a Texture Image.”

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Then line 158 sets u_Clicked to false and redraws the cube at line 159. This turns the
cube back to its original color. Line 161 returns picked as the return value.
Note, if at this point you call any function that returns control to the browser, such as
alert(), the content of the color buffer will be displayed on the  at that point.
For example, if you execute alert('The cube was selected!') at line 156, the red cube

will be displayed when you click the cube.
This approach, although simple, can handle more than one object by assigning different colors to each object. For example, red, blue, and green are enough if there are three
objects. For larger numbers of objects, you can use individual bits. Because there are 8 bits
for each component in RGBA, you can represent 255 objects just by using the R component. However, if the 3D objects are complex or the drawing area is large, it will take
some time to process the selection of objects. To overcome this disadvantage, you can use
simplified models to select objects or shrink the drawing area. In such cases, you can use
the framebuffer object, which will be explained in the section “Use What You’ve Drawn as
a Texture Image” later in this chapter.

Select the Face of the Object
You can also apply the method explained in the previous section to select a particular
face of an object. Let’s customize PickObject to build PickFace, a program that turns the
selected face white. Figure 10.4 shows PickFace.

Click

Figure 10.4

PickFace

PickFace is easy once you understand how PickObject works. PickObject drew the cube

in red when the mouse was clicked, resulting in the object’s display area in the color
buffer being red. By reading the pixel value of the clicked point and seeing if the color
of the pixel at the position was red, the program could determine if the object had been
selected. PickFace goes one step further and inserts the information of which face has
been selected into the color buffer. Here, you will insert the information in the alpha
component of the RGBA value. Let’s take a look at the sample program.

Select an Object

365

Sample Program (PickFace.js)
PickFace.js is shown in Listing 10.3. Some parts, such as the fragment shader, are

omitted for brevity.
Listing 10.3
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
33
50
51
74
75
76
77
78
79
80
81
82
83
84
85
86
87
366

PickFace.js

// PickFace.js
// Vertex shader program
var VSHADER_SOURCE =
'attribute vec4 a_Position;\n' +
'attribute vec4 a_Color;\n' +
'attribute float a_Face;\n' +
// Surface number (Cannot use int)
'uniform mat4 u_MvpMatrix;\n' +
'uniform int u_PickedFace;\n' + // Surface number of selected face
'varying vec4 v_Color;\n' +
'void main() {\n' +
' gl_Position = u_MvpMatrix * a_Position;\n' +
' int face = int(a_Face);\n' + // Convert to int
' vec3 color = (face == u_PickedFace) ? vec3(1.0):a_Color.rgb;\n'+
' if(u_PickedFace == 0) {\n' + // Insert face number into alpha
'
v_Color = vec4(color, a_Face/255.0);\n' +
' } else {\n' +
'
v_Color = vec4(color, a_Color.a);\n' +
' }\n' +
'}\n';
...
function main() {
...
// Set vertex information
var n = initVertexBuffers(gl);
...
// Initialize selected surface
gl.uniform1i(u_PickedFace, -1);
var currentAngle = 0.0; // Current rotation angle (degrees)
// Register event handlers
canvas.onmousedown = function(ev) {
// Mouse is pressed
var x = ev.clientX, y = ev.clientY;
var rect = ev.target.getBoundingClientRect();
if (rect.left <= x && x < rect.right && rect.top <= y && y < rect.bottom) {
// If clicked position is inside the , update the face
var x_in_canvas = x - rect.left, y_in_canvas = rect.bottom - y;
var face = checkFace(gl, n, x_in_canvas, y_in_canvas,
➥currentAngle, u_PickedFace, viewProjMatrix, u_MvpMatrix);
gl.uniform1i(u_PickedFace, face); // Pass the surface number
draw(gl, n, currentAngle, viewProjMatrix, u_MvpMatrix);

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88
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127
128
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133
134
154

164
165
166
167
168
169
170
171
172
173
174

}
}
...
function initVertexBuffers(gl) {
...
var vertices = new Float32Array([
// Vertex coordinates
1.0, 1.0, 1.0, -1.0, 1.0, 1.0, -1.0,-1.0, 1.0,
1.0,-1.0, 1.0,
1.0, 1.0, 1.0,
1.0,-1.0, 1.0,
1.0,-1.0,-1.0,
1.0, 1.0,-1.0,
...
1.0,-1.0,-1.0, -1.0,-1.0,-1.0, -1.0, 1.0,-1.0,
1.0, 1.0,-1.0
]);
...
var faces = new Uint8Array([
// Surface number
1, 1, 1, 1,
// v0-v1-v2-v3 Front
2, 2, 2, 2,
// v0-v3-v4-v5 Right
...
6, 6, 6, 6,
// v4-v7-v6-v5 Depth
]);
...
if (!initArrayBuffer(gl, faces, gl.UNSIGNED_BYTE, 1,
➥'a_Face')) return -1; // Surface Information
...
}
function checkFace(gl, n, x, y, currentAngle, u_PickedFace, viewProjMatrix,
➥u_MvpMatrix) {
var pixels = new Uint8Array(4); // Array for storing the pixel
gl.uniform1i(u_PickedFace, 0); // Write surface number into alpha
draw(gl, n, currentAngle, viewProjMatrix, u_MvpMatrix);
// Read the pixels at (x, y). pixels[3] is the surface number
gl.readPixels(x, y, 1, 1, gl.RGBA, gl.UNSIGNED_BYTE, pixels);
return pixels[3];
}

Let’s take a look from the vertex shader. a_Face at line 6 is the attribute variable used to
pass the surface number, which is then “coded” into the alpha value when the mouse
is clicked. The surface numbers are set up in initVertexBuffers() defined at line 99
and simply map vertices to a surface. Lines 127 onward define these mappings. So, for
example, vertices v0-v1-v2-v3 define a surface that is numbered 1, vertices v0-v3-v4-v5 are
numbered 2, and so on. Because each vertex needs a number to pass to the vertex shader,
there are four 1s written at line 128 to represent the first face.
If a face is already selected, u_PickedFace at line 8 informs the vertex shader of the
selected face number, allowing the shader to switch the way it draws the face based on
this information.
Select an Object

367

Line 12 converts a_Face, the surface number that is a float type, into an int type because
an int type cannot be used in the attribute variables (Chapter 6, “The OpenGL ES Shading
Language [GLSL ES]”). If the selected surface number is the same as the surface number
currently being manipulated, white is assigned to color at line 13. Otherwise, the original
surface color is assigned. If the mouse has been clicked (that is, u_PickedFace is set to 0),
the a_Face value is inserted into the alpha value and the cube is drawn (line 15).
Now, by passing 0 into u_PickedFace when the mouse is clicked, the cube is drawn with
an alpha value set to the surface number. u_PickedFace is initialized to –1 at line 75.
There is no surface with the number –1 (refer to the faces array at line 127), so the cube is
initially drawn without surfaces selected.
Let’s take a look at the essential processing of the event handler. u_PickedFace is passed
as an argument to checkFace() at line 85, which returns the surface number of the picked
face, checkFace(), at line 166. At line 168, 0 is passed to u_PickedFace to tell the vertex
shader that the mouse has been clicked. When draw() is called in the next line, the
surface number is inserted into the alpha value and the object is redrawn. Line 171 checks
the pixel value of the clicked point, and line 173 retrieves the inserted surface number
by using pixels[3]. (It is the alpha value, so the subscript is 3.) This surface number is
returned to the main code and then used at lines 86 and 87 to draw the cube. The vertex
shader handles the rest of the processing, as described earlier.

HUD (Head Up Display)
The Head Up Display, originally developed for aircraft, is a transparent display that presents data without requiring users to look away from their usual viewpoints. A similar
effect can be achieved in 3D graphics and used to overlay textual information on the 3D
scene. Here, you will construct a sample program that will display a diagram and some
information on top of the 3D graphics (HUD), as you can see in Figure 10.5.

Figure 10.5
368

HUD

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The goal of the program is to draw a triangle and some simple information about the 3D
scene, including the current rotation angle of the cube (from PickObject) that will change
as the cube rotates.

How to Implement a HUD
This HUD effect can be implemented using HTML and the canvas function without
WebGL. This is done as follows:
1. In the HTML file, prepare a  to draw the 3D graphics using WebGL and
another  to draw the HUD using the canvas function. In other words,
prepare two  and place the HUD on top of the WebGL canvas.
2. Draw the 3D graphics using the WebGL API on the  for WebGL.
3. Draw the HUD using the canvas functions on the  for the HUD.
As you can see, this is extremely simple and shows the power of WebGL and its ability to
mix 2D and 3D graphics within the browser. Let’s take a look at the sample program.

Sample Program (HUD.html)
Because we need to make changes to the HTML file to add the extra canvas, we show HUD.
html in Listing 10.4, with the additions in bold.
Listing 10.4
1
2
8
9
10
11
12

18
19
20

HUD.html



...


Please use a browser that supports "canvas"


...




The style attribute, used to define how an element looks or how it is arranged, allows you
to place the HUD canvas on top of the WebGL canvas. Style information is composed of
the property name and the value separated with a : as seen at line 9: style="position:
absolute". Multiple style elements are separated with ;.

HUD (Head Up Display)

369

In this example, you use position, which specifies how the element is placed, and the
z-index, which specifies the hierarchical relationship.
You can specify the position of the element in the absolute coordinate if you use absolute
for the position value. Unless you specify the position, all the elements specified to this
attribute will be place at the same position. z-index specifies the order in which elements
are displayed when multiple elements are at the same position. The element with the
larger number will be displayed over the one with a smaller number. In this case, the
z-index of the  for the HUD at line 12 is 1.
The result of this code is two  elements, placed at the same location with
the  that displays the HUD on top of the  that displays the WebGL.
Conveniently, the background of the canvas element is transparent by default, so the
WebGL canvas can be seen through the HUD canvas. Anything that is drawn on the HUD
canvas will appear over the 3D objects and create the effect of a HUD.

Sample Program (HUD.js)
Next, let’s take a look at HUD.js in Listing 10.5. There are two changes made compared to
PickObject.js:
1. Retrieve the rendering context to draw in the  for the HUD and use it to
draw.
2. Register the event handler when the mouse is clicked to the  for the HUD
and not to the  for WebGL.
Step 1 simply uses the source code used in Chapter 2, “Your First Step with WebGL,” to
draw a triangle onto the . Step 2 is required to ensure that mouse click information is passed to the HUD canvas rather than the WebGL canvas. The vertex shader and
fragment shader are the same as PickObject.js.
Listing 10.5
1
30
31
32
33
40
41
42
43
82
83
370

HUD.js

// HUD.js
...
function main() {
// Retrieve  element
var canvas = document.getElementById('webgl');
var hud = document.getElementById('hud');
...
// Get the rendering context for WebGL
var gl = getWebGLContext(canvas);
// Get the rendering context for 2DCG
var ctx = hud.getContext('2d');
...
// Register the event handler
hud.onmousedown = function(ev) {
// Mouse is pressed

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...
89

check(gl, n, x_in_canvas, y_in_canvas, currentAngle, u_Clicked,
➥viewProjMatrix, u_MvpMatrix);
...
}

91
92
93
94
95
96
97
98
99
100

}

184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199

...
function draw2D(ctx, currentAngle) {
ctx.clearRect(0, 0, 400, 400);
// Clear 
// Draw triangle with white lines
ctx.beginPath();
// Start drawing
ctx.moveTo(120, 10); ctx.lineTo(200, 150); ctx.lineTo(40, 150);
ctx.closePath();
ctx.strokeStyle = 'rgba(255, 255, 255, 1)'; // Set the line color
ctx.stroke();
// Draw triangle with white lines
// Draw white letters
ctx.font = '18px "Times New Roman"';
ctx.fillStyle = 'rgba(255, 255, 255, 1)'; // Set the letter color
ctx.fillText('HUD: Head Up Display', 40, 180);
ctx.fillText('Triangle is drawn by Hud API.', 40, 200);
ctx.fillText('Cube is drawn by WebGL API.', 40, 220);
ctx.fillText('Current Angle: '+ Math.floor(currentAngle), 40, 240);
}

var tick = function() {
// Start drawing
currentAngle = animate(currentAngle);
draw2D(ctx, currentAngle); // Draw 2D
draw(gl, n, currentAngle, viewProjMatrix, u_MvpMatrix);
requestAnimationFrame(tick, canvas);
};
tick();

Because the processing flow of the program is straightforward, let’s take a look from
main() at line 30. First, line 33 obtains the  element for the HUD. This is used to
get the drawing context for the 2D graphics (Chapter 2) at line 43, which is used to draw
the HUD. You register the mouse-click event handler for the HUD canvas (hud) instead of
the WebGL canvas in PickObject.js. This is because the event goes to the HUD canvas,
which is placed on top of the WebGL canvas.
The code from line 93 handles the animation and uses draw2D(), added at line 95, to draw
the HUD information.
draw2D() is defined at line 184 and takes ctx parameters, the context to draw on the
canvas, and the current rotation angle, currentAngle. Line 185 clears the HUD canvas
using the clearRect() method, which takes the upper-left corner, the width, and the
height of the rectangle to clear. Lines 187 to 191 draw the triangle which, unlike drawing

HUD (Head Up Display)

371

a rectangle as explained in Chapter 2, requires that you define the path (outline) of a
triangle to draw it. Lines 187 to 191 define the path, set the color, and draw the triangle.
Lines 193 onward specify the text color and font and then use fillText(), which specifies the letters to draw as the first parameter and the x and y coordinates to draw as the
second and third parameters, to actually write the text. Line 198 displays the current
rotation angle and uses Math.floor() to truncate the numbers below the decimal point.
Line 185 clears the canvas because the displayed value (rotation angle) changes at each
drawing.

Display a 3D Object on a Web Page (3DoverWeb)
Displaying a 3D object on a web page is simple with WebGL and the inverse of the HUD
example. In this case, the WebGL canvas is on top of the web page, and the canvas is set
to transparent. Figure 10.6 shows 3DoverWeb.

Figure 10.6

3DoverWeb2

3DoverWeb.js is based on PickObject.js with almost no changes. The only change is that
the alpha value of the clear color is changed from 1.0 to 0.0 at line 55.
55

gl.clearColor(0.0, 0.0, 0.0, 0.0);

By making the alpha value 0.0, the background of the WebGL canvas becomes transparent, and you can see the web page behind the WebGL . You can also experiment
with the alpha value; any value other than 1.0 changes the transparency and makes the
web page more or less visible.

Fog (Atmospheric Effect)
In 3D graphics, the term fog is used to describe the effect that makes a distant object seem
hazy. The term describes objects in any medium, so objects underwater can also have a
2

The sentences on the web page on the background are from the book The Design of Design (by
Frederick P. Brooks Jr, Pearson).

372

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fog effect applied. Here, you construct a sample program Fog that realizes the fog effect.
Figure 10.7 shows a screen shot. You can adjust the density of the fog with the up/down
arrow keys. Try running the sample program and experiment with the effect.

Figure 10.7

Fog

How to Implement Fog
There are various ways to calculate fog, but here you will use a linear computation (linear
fog) because the calculation is easy. The linear fog method determines the density of the
fog by setting the starting point (the distance where the object starts to become hazy) and
the end point (where the object is completely obscured). The density of the fog between
these points is changed linearly. Note that the end point is not where the fog ends; rather,
it is where the fog becomes so dense that it obscures all objects. We will call how clearly
we can see the object the fog factor; it is calculated, in the case of linear fog, as follows:
Equation 10.1
fog factor =

( end point − distance from eye point )
( end point − starting point )

Where
starting point ≤ distance from eye point ≤ end point
When the fog factor is 1.0, you can see the object completely, and if it 0.0, you cannot see
it at all (see Figure 10.8). The fog factor is 1.0 when the (distance from eye point)
< (starting point), and 0.0 when (end point) < (distance from eye point).

Fog (Atmospheric Effect)

373

fog factor
1.0

eye point
distance from
eye point
start distance

Figure 10.8

end distance

Fog factor

You can calculate the color of a fragment based on the fog factor, as follows in Equation
10.2.
Equation 10.2
fragment color =

surface color × fog factor + fog color × (1 − fog factor

)

Now, let’s take a look at the sample program.

Sample Program (Fog.js)
The sample program is shown in Listing 10.6. Here, you (1) calculate the distance of the
object (vertex) from the eye point in the vertex shader, and based on that, you (2) calculate the fog factor and the color of the object based on the fog factor in the fragment
shader. Note that this program specifies the position of the eye point with the world
coordinate system (see Appendix G, “World Coordinate System Versus Local Coordinate
System”) so the fog calculation takes place in the world coordinate system.
Listing 10.6
1
2
3
7
8
9
10
11
12
13
14
15
16
17
374

Fog.js

// Fog.js
// Vertex shader program
var VSHADER_SOURCE =
...
'uniform mat4 u_ModelMatrix;\n' +
'uniform vec4 u_Eye;\n' + // The eye point (world coordinates)
'varying vec4 v_Color;\n' +
'varying float v_Dist;\n' +
'void main() {\n' +
' gl_Position = u_MvpMatrix * a_Position;\n' +
' v_Color = a_Color;\n' +
// Calculate the distance to each vertex from eye point
' v_Dist = distance(u_ModelMatrix * a_Position, u_Eye);\n' +
'}\n';

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<-(1)

18
19
23
24
25
26
27
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29
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31
32
33
34
35
53
59
60
61
62
63
64
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77
78
79
80
81
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93
97

// Fragment shader program
var FSHADER_SOURCE =
...
'uniform vec3 u_FogColor;\n' + // Color of Fog
'uniform vec2 u_FogDist;\n' + // Fog starting point, end point)
'varying vec4 v_Color;\n' +
'varying float v_Dist;\n' +
'void main() {\n' +
// Calculate the fog factor
<-(2)
' float fogFactor = clamp((u_FogDist.y - v_Dist) / (u_FogDist.y –
➥u_FogDist.x), 0.0, 1.0);\n' +
// u_FogColor * (1 - fogFactor) + v_Color * fogFactor
' vec3 color = mix(u_FogColor, vec3(v_Color), fogFactor);\n' +
' gl_FragColor = vec4(color, v_Color.a);\n' +
'}\n';
function main() {
...
var n = initVertexBuffers(gl);
...
// Color of fog
var fogColor = new Float32Array([0.137, 0.231, 0.423]);
// Distance of fog [fog starts, fog completely covers object]
var fogDist = new Float32Array([55, 80]);
// Position of eye point (world coordinates)
var eye = new Float32Array([25, 65, 35]);
...
// Pass fog color, distances, and eye point to uniform variable
gl.uniform3fv(u_FogColor, fogColor); // Fog color
gl.uniform2fv(u_FogDist, fogDist);
// Starting point and end point
gl.uniform4fv(u_Eye, eye);
// Eye point
// Set clear color and enable hidden surface removal function
gl.clearColor(fogColor[0], fogColor[1], fogColor[2], 1.0);
...
mvpMatrix.lookAt(eye[0], eye[1], eye[2], 0, 2, 0, 0, 1, 0);
...
document.onkeydown = function(ev){ keydown(ev, gl, n, u_FogDist, fogDist); };
...

The calculation of the distance from the eye point to the vertex, done by the vertex
shader, is straightforward. You simply transform the vertex coordinates to the world coordinates using the model matrix and then call the built-in function distance() with the
position of the eye point (world coordinates) and the vertex coordinates. The distance()

Fog (Atmospheric Effect)

375

function calculates the distance between two coordinates specified by the arguments. This
calculation takes place at line 15, and the result is then written to the v_Dist variable and
passed to the fragment shader.
The fragment shader calculates the fogged color of the object using Equations 10.1 and
10.2. The fog color, fog starting point, and fog end point, which are needed to calculate
the fogged color, are passed in the uniform variables u_FogColor and u_FogDist at lines 23
and 24. u_FogDist.x is the starting point, and u_FogDist.y is the end point.
The fog factor is calculated at line 29 using Equation 10.1. The clamp() function is a builtin function; if the value specified by the first parameter is outside the range specified by
the second and third parameter ([0.0 0.1] in this case), it will fix the value to one within
the range. In other words, the value is fixed to 0.0 if the value is smaller than 0.0, and 1.0
if the value is larger than 1.0. If the value is within the range, the value is unchanged.
Line 31 is the calculation of the fragment color using the fog factor. This implements
Equation 10.2 and uses a built-in function, mix(), which calculates x*(1–z)+y*z, where x is
the first parameter, y is the second, and z is the third.
The processing in JavaScript’s main() function from line 35 sets up the values necessary
for calculating the fog in the appropriate uniform variables.
You should note that there are many types of fog calculations other than linear fog, for
example exponential fog, used in OpenGL (see the book OpenGL Programming Guide). You
can implement these fog calculations using the same approach, just changing the calculation method in the fragment shader.

Use the w Value (Fog_w.js)
Because the distance calculation within the shader can affect performance, an alternative
method allows you to easily approximate the calculation of the distance from the eye
point to the object (vertex) by using the w value of coordinates transformed by the model
view projection conversion. In this case, the coordinates are substituted in gl_Position.
The fourth component, w of gl_Position which you haven’t used explicitly before, is the
z value of each vertex in the view coordinate system multiplied by –1. The eye point is the
origin in the view coordinates, and the view direction is the negative direction of z, so z
is a negative value. The w value, which is the z value multiplied by –1, can be used as an
approximation of the distance.
If you reimplement the calculation in the vertex shader using w, as shown in Listing 10.7,
the fog effect will work as before.
Listing 10.7
1
2
3

376

Fog_w.js

// Fog_w.js
// Vertex shader program
var VSHADER_SOURCE =
...

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7
8
9
10
11
12
13
14

'varying vec4 v_Color;\n' +
'varying float v_Dist;\n' +
'void main() {\n' +
' gl_Position = u_MvpMatrix * a_Position;\n' +
' v_Color = a_Color;\n' +
// Use the negative z value of vertex in view coordinate system
' v_Dist = gl_Position.w;\n' +
'}\n';

Make a Rounded Point
In Chapter 2, you constructed a sample program that draws a point to help you understand the basics of shaders. However, to allow you to focus on the operation of the
shaders, the point displayed wasn’t “round” but actually “square,” which is simpler to
draw. In this section, you construct a sample program, RoundedPoint, which draws a
round point (see Figure 10.9).

Figure 10.9

A screen shot of RoundedPoint

How to Implement a Rounded Point
To draw a “round” point, you just have to make the “rectangle” point round. This can be
achieved using the rasterization process that takes place between the vertex shader and
the fragment shader and was explained in Chapter 5, “Using Colors and Texture Images.”
This rasterization process generates a rectangle consisting of multiple fragments, and each
fragment is passed to the fragment shader. If you draw these fragments as-is, a rectangle
will be displayed. So you just need to modify the fragment shader to draw only the fragments inside the circle, as shown in Figure 10.10.

Make a Rounded Point

377

fragments to be discarded

Figure 10.10

Discarding fragments to turn a rectangle into a circle

To achieve this, you need to know the position of each fragment created during rasterization. In Chapter 5, you saw a sample program that uses the built-in variable gl_FragCoord
to pass (input) the data to the fragment shader. In addition to this, there is one more
built-in variable gl_PointCoord, which is suitable for drawing a round point (see Table
10.1).
Table 10.1

Built-In Variables of Fragment Shader (Input)

Type and Name of Variable

Description

vec4 gl_FragCoord

Window coordinates of fragment

vec4 gl_PointCoord

Position of fragment in the drawn point (0.0 to 1.0)

gl_PointCoord gives the position of each fragment taken from the range (0.0, 0.0) to (1.0,

1.0), as shown in Figure 10.11. To make the rectangle round, you simply have to discard
the fragments outside the circle centered at (0.5, 0.5) with radius 0.5. You can use the
discard statement to discard these fragments.
(0.0, 0.0)
(0.5, 0.5)

0.5

Figure 10.11

(1.0, 1.0)

Coordinates of gl_PointCoord

Sample Program (RoundedPoints.js)
The sample program is shown in Listing 10.8. This is derived from MultiPoint.js, which
was used in Chapter 4, “More Transformations and Basic Animation,” to draw multiple
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points. The only difference is in the fragment shader. The vertex shader is also shown for
reference.
Listing 10.8
1
2
3
4
5
6
7
8
9
10
11
15
16
17
18
19
20
21
22
53
54

RoundedPoint.js

// RoundedPoints.js
// Vertex shader program
var VSHADER_SOURCE =
'attribute vec4 a_Position;\n' +
'void main() {\n' +
' gl_Position = a_Position;\n' +
' gl_PointSize = 10.0;\n' +
'}\n';
// Fragment shader program
var FSHADER_SOURCE =
...
'void main() {\n' +
// Center coordinate is (0.5, 0.5)
' float dist = distance(gl_PointCoord, vec2(0.5, 0.5));\n' +
' if(dist < 0.5) {\n' +
// Radius is 0.5
'
gl_FragColor = vec4(1.0, 0.0, 0.0, 1.0);\n' +
' } else { discard; }\n' +
'}\n';
function main() {
...
gl.drawArrays(gl.POINTS, 0, n);
}

The key difference is the calculation, starting at line 16, which determines whether a fragment should be discarded. gl_PointCoord holds the fragment’s position (specified in the
range 0.0 to 0.1), and the center point is (0.5, 0.5). Therefore, to make a rectangle point
round, you have to do the following:
1. Calculate the distance from the center (0.5, 0.5) to each fragment.
2. Draw those fragments for which the distance is less than 0.5.
In RoundedPoint.js, the distance calculation takes place at line 16. Here, you just have to
calculate the distance between the center point (0.5, 0.5) and gl_PointCoord. Because the
gl_PointCoord is a vec2 type, you need to pass (0.5, 0.5) to distance() as a vec2.
Once you have calculated the distance from the center, it is used at line 17 to check
whether the distance is less than 0.5 (in other words, whether the fragment is in the
circle). If the fragment is in circle, the fragment is drawn so line 18 uses gl_FragColor to
set the draw color. Otherwise, at line 19, the discard statement causes WebGL to automatically throw away the fragment.
Make a Rounded Point

379

Alpha Blending
The alpha value controls the transparency of drawn objects. If you specify 0.5 as the alpha
value, the object becomes semi-transparent, allowing anything drawn underneath it to be
partially visible. As the alpha value approaches 0, more of the background objects appear.
If you try this yourself, you’ll actually see that as you decrease the alpha value, WebGL
objects become white. This is because WebGL’s default behavior is to use the same alpha
value for both objects and the . In the sample programs, the web page behind
the  is white, so this shows through.
Let’s construct a sample program that shows how to use alpha blending to get the desired
effect. The function that allows the use of the alpha value is called an alpha blending (or
simply blending) function. This function is already built into WebGL, so you just need to
enable it to tell WebGL to start to use the alpha values supplied.

How to Implement Alpha Blending
You’ll need the following two steps to enable and use the alpha blending function.
1. Enable the alpha blending function:
gl.enable(gl.BLEND);

2. Specify the blending function:
gl.blendFunc(gl.SRC_ALPHA, gl.ONE_MINUS_SRC_ALPHA);

The blending function will be explained later, so let’s try using the sample program. Here,
we will reuse LookAtTrianglesWithKey_ViewVolume described in Chapter 7, “Toward the
3D World.” As shown in Figure 10.12, this program draws three triangles and allows the
position of the eye point to be changed using the arrow key.

Figure 10.12
380

A screen shot of LookAtTrianglesWithKeys_ViewVolume

CHAPTER 10

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Let’s add the code for steps 1 and 2, specify 0.4 as the alpha value of the color of the
triangles, and call the resulting program LookAtBlendedTriangles. Figure 10.13 shows the
effect when run. As you can see, all triangles became semitransparent, and you are able to
see the triangles behind. When you move the eye point with the arrow key, you can see
that the blending is continuously taking place.

Figure 10.13

A screen shot of LookAtBlendedTriangles

Let’s look at the sample program.

Sample Program (LookAtBlendedTriangles.js)
LookAtBlendedTriangles.js is shown in Listing 10.9. The code that has changed is in

lines 51 to 54, and the alpha value (0.4) is added to the definition of color information
in initVertexBuffer() at lines 81 to 91. Accordingly, the size and stride parameters have
changed for gl.vertexAttribPointer().
Listing 10.9
1
2
25
43
51
52
53
54

LookAtBlenderTriangles.js

// LookAtBlendedTriangles.js
// LookAtTrianglesWithKey_ViewVolume.js is the original
...
function main() {
...
var n = initVertexBuffers(gl);
...
// Enable alpha blending
gl.enable (gl.BLEND);
// Set blending function
gl.blendFunc(gl.SRC_ALPHA, gl.ONE_MINUS_SRC_ALPHA);
...

Alpha Blending

381

75
76
77
78
79
80
81
82
91
92
93
127
128

draw(gl, n, u_ViewMatrix, viewMatrix);
}
function initVertexBuffers(gl) {
var verticesColors = new Float32Array([
// Vertex coordinates and color(RGBA)
0.0, 0.5, -0.4,
0.4, 1.0, 0.4, 0.4,
-0.5, -0.5, -0.4,
0.4, 1.0, 0.4, 0.4,
...
0.5, -0.5, 0.0,
1.0, 0.4, 0.4, 0.4,
]);
var n = 9;
...
return n;
}

Blending Function
Let’s explore the blending function gl.blendFunc() to understand how this can be used
to achieve the blending effect. You need two colors for blending: the color to blend
(source color) and the color to be blended (destination color). For example, when you
draw one triangle on top of the other, the color of the triangle already drawn is the destination color, and the color of the triangle drawn on top is the source color.
gl.blendFunc(src_factor, dst_factor)
Specify the method to blend the source color and the destination color. The blended
color is calculated as follows:

color ( RGB ) = source color × src_factor + destination color × dst_factor
Parameters

src_factor

Specifies the multiplier for the source color (Table 10.2).

dst_factor

Specifies the multiplier for the destination color (Table
10.2).

Return value

None

Errors

INVALID_ENUM

Table 10.2

src_factor and dst_factor are none of the values in Table
10.2

Constant Values that Can Be Specified as src_factor and dst_factor3

Constant

Multiplicand for R

Multiplicand for G

Multiplicand for B

gl.ZERO

0.0

0.0

0.0

gl.ONE

1.0

1.0

1.0

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Constant

Multiplicand for R

Multiplicand for G

Multiplicand for B

gl.SRC_COLOR

Rs

Gs

Bs

gl.ONE_MINUS_SRC_COLOR

(1 – Rs)

(1 – Gs)

(1 – Bs)

gl.DST_COLOR

Rd

Gd

Bd

gl.ONE_MINUS_DST_COLOR

(1 – Rd)

(1 – Bd)

(1 – Gd)

gl.SRC_ALPHA

As

As

As

gl.ONE_MINUS_SRC_ALPHA

(1 – As)

(1 – As)

(1 – As)

gl.DST_ALPHA

Ad

Ad

Ad

gl.ONE_MINUS_DST_ALPHA

(1 – Ad)

(1 – Ad)

(1 – Ad)

gl.SRC_ALPHA_SATURATE

min(As, Ad)

min(As, Ad)

min(As, Ad)

3
gl.CONSTANT_COLOR, gl.ONE_MINUSCONSTANT_COLOR, gl.CONSTANT_ALPHA, and gl.ONE_
MINUS_CONSTANT_ALPHA are removed from OpenGL.

(Rs,Gs,Bs,As) is the source color and (Rd,Gd,Bd,Ad) is the destination color.
In the sample program, you used the following:
54

gl.blendFunc(gl.SRC_ALPHA, gl.ONE_MINUS_SRC_ALPHA);

For example, if the source color is semitransparent green (0.0, 1.0, 0.0, 0.4) and the destination color is yellow (1.0, 1.0, 0.0, 1.0), src_factor becomes the alpha value 0.4 and dst_
factor becomes (1 – 0.4)=0.6. The calculation is shown in Figure 10.14.
c o lo r(R G B ) = s ource colo r * src_factor + destin ation co lor * dst_factor

*

R G B
source color 0.0 1.0 0.0
src_factor
0.4 0.4 0.4
0.0 0.4 0.0

*

destination color
dst_factor

Blended color

Figure 10.14

R G B
1.0 1.0 0.0
0.6 0.6 0.6
0.6 0.6 0.0

0.6 1.0 0.0

Calculation of gl.blendFunc(gl.SRC_ALPHA, gl.ONE_MINUS_SRC_ALPHA)

You can experiment with the other possible parameter values for src_factor, dst_factor, but
one that is often used is additive blending. When used, the result will become brighter
than the original value because it is a simple addition. It can be used for an indicator or
the lighting effect resulting from an explosion.
glBlendFunc(GL_SRC_ALPHA, GL_ONE);

Alpha Blending

383

Alpha Blend 3D Objects (BlendedCube.js)
Let’s now explore the effects of alpha blending on a representative 3D object, a cube, by
making it semitransparent. You will reuse the ColoredCube sample program from Chapter
7 to create BlendedCube, which adds the two steps needed for blending (see Listing 10.10).
Listing 10.10
1
47
48
49
50
51
52
53

BlendedCube.js

// BlendedCube.js
...
// Set the clear color and enable the depth test
gl.clearColor(0.0, 0.0, 0.0, 1.0);
gl.enable(gl.DEPTH_TEST);
// Enable alpha blending
gl.enable (gl.BLEND);
// Set blending function
gl.blendFunc(gl.SRC_ALPHA, gl.ONE_MINUS_SRC_ALPHA);

Unfortunately, if you run this program as-is, you won’t see the expected result (right side
of Figure 10.15); rather, you will see something similar to the left side, which is no different from the original ColoredCube used in Chapter 7.

Figure 10.15

BlendedCube

This is because of the hidden surface removal function enabled at line 49. Blending only
takes place on the drawn surfaces. When the hidden surface removal function is enabled,
the hidden surfaces are not drawn, so there is no other surface to be blended with.
Therefore, you don’t see the blending effect as expected. To solve this problem, you can
simply comment out line 49 that enables the hidden surface removal function.

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48
49
50
51

gl.clearColor(0.0, 0.0, 0.0, 1.0);
// gl.enable(gl.DEPTH_TEST);
// Enable alpha blending
gl.enable (gl.BLEND);

How to Draw When Alpha Values Coexist
This is a quick solution, but it’s not very satisfactory because, as we’ve seen in Chapter 7,
hidden surface removal is often needed to correctly draw a 3D scene.
You can overcome this problem by drawing objects while turning the hidden surface
removal function on and off.
1. Enable the hidden surface removal function.
gl.enable(gl.DEPTH_TEST);

2. Draw all the opaque objects (whose alpha values are 1.0).
3. Make the depth buffer (Chapter 7), which is used in the hidden surface removal,
read-only.
gl.depthMask(false);

4. Draw all the transparent objects (whose alpha values are smaller than 1.0). Note,
they should be sorted by the depth order and drawn back to front.
5. Make the depth buffer readable and writable.
gl.depthMask(true);

If you completely disable the hidden surface removal function, when there are transparent objects behind opaque objects, the transparent object will not be hidden behind the
opaque objects. So you need to control that with gl.depthMask(). gl.depthMask() has the
following specification.

gl.depthMask(mask)
Enable or disable writing into the depth buffer.
Parameters

mask

Return value

None

Errors

None

Specifies whether the depth buffer is enabled for writing. If mask
is false, depth buffer writing is disabled.

Alpha Blending

385

The depth buffer was briefly introduced in Chapter 7. The z values of fragments (which
are normalized to a value between 0.0 and 1.0) are written into the buffer. For example,
say there are two triangles on top of each other and you draw from the triangle on top.
First, the z value of the triangle on top is written into the depth buffer. Then, when the
triangle on bottom is drawn, the hidden surface removal function compares the z value
of its fragment that is going to be drawn, with the z value already written in the depth
buffer. Then only when the z value of the fragment that is going to be drawn is smaller
than the existing value in the buffer (that is, when it’s closer to the eye point) will the
fragment be drawn into the color buffer. This approach ensures that hidden surface
removal is achieved. Therefore, after drawing, the z value of the fragment of the surface
that can be seen from the eye point is left in the depth buffer.
Opaque objects are drawn into the color buffer in the correct order by removing the
hidden surfaces in the processing of steps 1 and 2, and the z value that represents the
order is written in the depth buffer. Transparent objects are drawn into the color buffer
using that z value in steps 3, 4, and 5, so the hidden surfaces of the transparent objects
behind the opaque objects will be removed. This results in the correct image being shown
where both objects coexist.

Switching Shaders
The sample programs in this book draw using a single vertex shader and a single fragment
shader. If all objects can be drawn with the same shaders, there is no problem. However,
if you want to change the drawing method for each object, you need to add significant
complexity to the shaders to achieve multiple effects. A solution is to prepare more than
one shader and then switch between these shaders as required. Here, you construct a
sample program, ProgramObject, which draws a cube colored with a single color and
another cube with a texture image. Figure 10.16 shows a screen shot.

Figure 10.16

386

A screen shot of ProgramObject

CHAPTER 10

Advanced Techniques

This program is also an example of the shading of an object with a texture image.

How to Implement Switching Shaders
The shaders can be switched easily by creating program objects, as explained in Chapter 8,
“Lighting Objects,” and switching them before drawing. Switching is carried out using the
function gl.useProgram(). Because you are explicitly manipulating shader objects, you
cannot use the convenience function initShaders(). However, you can use the function
createProgram() in cuon-utils.js, which is called from initShaders().
The following is the processing flow of the sample program. It performs the same procedure twice, so it looks long, but the essential code is simple:
1. Prepare the shaders to draw an object shaded with a single color.
2. Prepare the shaders to draw an object with a texture image.
3. Create a program object that has the shaders from step 1 with createProgram().
4. Create a program object that has the shaders from step 2 with createProgram().
5. Specify the program object created by step 3 with gl.useProgram().
6. Enable the buffer object after assigning it to the attribute variables.
7. Draw a cube (drawn in a single color).
8. Specify the program object created in step 4 using gl.useProgram().
9. Enable the buffer object after assigning it to the attribute variables.
10. Draw a cube (texture is pasted).
Now let’s look at the sample program.

Sample Program (ProgramObject.js)
The key program code for steps 1 to 4 is shown in Listing 10.11. Two types of vertex
shader and fragment shader are prepared: SOLID_VSHADER_SOURCE (line 3) and SOLID_
FSHADER_SOURCE (line 19) to draw an object in a single color, and TEXTURE_VSHADER_SOURCE
(line 29) and TEXTURE_FSHADER_SOURCE (line 46) to draw an object with a texture image.
Because the focus here is on how to switch the program objects, the contents of the
shaders are omitted.
Listing 10.11
1
2
3
18

ProgramObject (Process for Steps 1 to 4)

// ProgramObject.js
// Vertex shader for single color drawing
var SOLID_VSHADER_SOURCE =
...
// Fragment shader for single color drawing

<- (1)

Switching Shaders

387

19
28
29
45
46
58
69
70
71

77
78
79
83
84
85
89
99
100
106
107
122
123
124
125
128
129
130
131

var SOLID_FSHADER_SOURCE =
...
// Vertex shader for texture drawing
<- (2)
var TEXTURE_VSHADER_SOURCE =
...
// Fragment shader for texture drawing
var TEXTURE_FSHADER_SOURCE =
...
function main() {
...
// Initialize shaders
var solidProgram = createProgram(gl, SOLID_VSHADER_SOURCE,
➥SOLID_FSHADER_SOURCE);
<- (3)
var texProgram = createProgram(gl, TEXTURE_VSHADER_SOURCE,
➥TEXTURE_FSHADER_SOURCE); <- (4)
...
// Get the variables in the program object for single color drawing
solidProgram.a_Position = gl.getAttribLocation(solidProgram, 'a_Position');
solidProgram.a_Normal = gl.getAttribLocation(solidProgram, 'a_Normal');
...
// Get the storage location of attribute/uniform variables
texProgram.a_Position = gl.getAttribLocation(texProgram, 'a_Position');
texProgram.a_Normal = gl.getAttribLocation(texProgram, 'a_Normal');
...
texProgram.u_Sampler = gl.getUniformLocation(texProgram, 'u_Sampler');
...
// Set vertex information
var cube = initVertexBuffers(gl, solidProgram);
...
// Set texture
var texture = initTextures(gl, texProgram);
...
// Start drawing
var currentAngle = 0.0; // Current rotation angle (degrees)
var tick = function() {
currentAngle = animate(currentAngle); // Update rotation angle
...
// Draw a cube in single color
drawSolidCube(gl, solidProgram, cube, -2.0, currentAngle, viewProjMatrix);
// Draw a cube with texture
drawTexCube(gl, texProgram, cube, texture, 2.0, currentAngle,
➥viewProjMatrix);

132
133

388

window.requestAnimationFrame(tick, canvas);

CHAPTER 10

Advanced Techniques

134
135
136
137
138
148
149
150
154
155
156
157

};
tick();
}
function initVertexBuffers(gl, program) {
...
var vertices = new Float32Array([
// Vertex coordinates
1.0, 1.0, 1.0, -1.0, 1.0, 1.0, -1.0,-1.0, 1.0, 1.0,-1.0, 1.0,
1.0, 1.0, 1.0, 1.0,-1.0, 1.0, 1.0,-1.0,-1.0, 1.0, 1.0,-1.0,
...
1.0,-1.0,-1.0, -1.0,-1.0,-1.0, -1.0, 1.0,-1.0, 1.0, 1.0,-1.0
]);
var normals = new Float32Array([
...
]);

164
165
166

var texCoords = new Float32Array([ // Texture coordinates
...
]);

173
174
175

var indices = new Uint8Array([
...
]);

182
183
184
185
186
187
188
189
190

// Indices for vertices

var o = new Object(); // Use Object to return buffer objects
// Write vertex information to buffer object
o.vertexBuffer = initArrayBufferForLaterUse(gl, vertices, 3, gl.FLOAT);
o.normalBuffer = initArrayBufferForLaterUse(gl, normals, 3, gl.FLOAT);
o.texCoordBuffer = initArrayBufferForLaterUse(gl, texCoords, 2, gl.FLOAT);
o.indexBuffer = initElementArrayBufferForLaterUse(gl, indices,
➥gl.UNSIGNED_BYTE);
...
o.numIndices = indices.length;
...
return o;

193
199
200

// Normal

}

Starting with the main() function in JavaScript, you first create a program object for
each shader with createProgram() at lines 70 and 71. The arguments of the createProgram() are the same as the initShaders(), and the return value is the program object.
You save each program object in solidProgram and texProgram. Then you retrieve the
storage location of the attribute and uniform variables for each shader at lines 78 to 89.
You will store them in the corresponding properties of the program object, as you did in
Switching Shaders

389

MultiJointModel_segment.js. Again, you leverage JavaScript’s ability to freely append a

new property of any type to an object.
The vertex information is then stored in the buffer object by initVertexBuffers() at line
100. You need (1) vertex coordinates, (2) the normals, and (3) indices for the shader to
draw objects in a single color. In addition, for the shader to draw objects with a texture
image, you need the texture coordinates. The processing in initVertexBuffers() handles
this and binds the correct buffer object to the corresponding attribute variables when the
program object is switched.
initVertexBuffers() prepares the vertex coordinates from line 148, normals from line

157, texture coordinates from line 166, and index arrays from line 175. Line 184 creates
object (o) of type Object. Then you store the buffer object to the property of the object
(lines 187 to 190). You can maintain each buffer object as a global variable, but that introduces too many variables and makes it hard to understand the program. By using properties, you can more conveniently manage all four buffer objects using one object o.4
You use initArrayBufferForLaterUse(), explained in MultiJointModel_segment.js, to
create each buffer object. This function writes vertex information into the buffer object
but does not assign it to the attribute variables. You use the buffer object name as its
property name to make it easier to understand. Line 199 returns the object o as the return
value.
Once back in main() in JavaScript, the texture image is set up in initTextures() at line
107, and then everything is ready to allow you to draw the two cube objects. First, you
draw a single color cube using drawSolidCube() at line 129, and then you draw a cube
with a texture image by using drawTexCube() at line 131. Listing 10.12 shows the latter
half of the steps, steps 5 through 10.
Listing 10.12
236
237
238
239
240
241
242
243
244
245
246
247

4

ProgramObject.js (Processes for Steps 5 through 10)

function drawSolidCube(gl, program, o, x, angle, viewProjMatrix) {
gl.useProgram(program);
// Tell this program object is used
// Assign the buffer objects and enable the assignment
initAttributeVariable(gl, program.a_Position, o.vertexBuffer);
initAttributeVariable(gl, program.a_Normal, o.normalBuffer);
gl.bindBuffer(gl.ELEMENT_ARRAY_BUFFER, o.indexBuffer);
drawCube(gl, program, o, x, angle, viewProjMatrix);

// Draw

<-(5)
<-(6)

<-(7)

}
function drawTexCube(gl, program, o, texture, x, angle, viewProjMatrix) {

To keep the explanation simple, the object (o) was used. However, it is better programming practice
to create a new user-defined type for managing the information about a buffer object and to use it to
manage the four buffers.

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248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
275
276
281
286
291
292

gl.useProgram(program);

// Tell this program object is used <-(8)

// Assign the buffer objects and enable the assignment
<-(9)
initAttributeVariable(gl, program.a_Position, o.vertexBuffer);
initAttributeVariable(gl, program.a_Normal, o.normalBuffer);
initAttributeVariable(gl, program.a_TexCoord, o.texCoordBuffer);
gl.bindBuffer(gl.ELEMENT_ARRAY_BUFFER, o.indexBuffer);
// Bind texture object to texture unit 0
gl.activeTexture(gl.TEXTURE0);
gl.bindTexture(gl.TEXTURE_2D, texture);
drawCube(gl, program, o, x, angle, viewProjMatrix); // Draw

<-(10)

}
// Assign the buffer objects and enable the assignment
function initAttributeVariable(gl, a_attribute, buffer) {
gl.bindBuffer(gl.ARRAY_BUFFER, buffer);
gl.vertexAttribPointer(a_attribute, buffer.num, buffer.type, false, 0, 0);
gl.enableVertexAttribArray(a_attribute);
}
...
function drawCube(gl, program, o, x, angle, viewProjMatrix) {
// Calculate a model matrix
...
// Calculate transformation matrix for normal
...
// Calculate a model view projection matrix
...
gl.drawElements(gl.TRIANGLES, o.numIndices, o.indexBuffer.type, 0);
}

drawSolidCube() is defined at line 236 and uses gl.useProgram() at line 237 to tell
the WebGL system that you will use the program (program object, solidProgram)
specified by the argument. Then you can draw using solidProgram. The buffer objects

for vertex coordinates and normals are assigned to attribute variables and enabled by
initAttributeVariable() at lines 240 and 241. This function is defined at line 264. Line
242 binds the buffer object for the indices to gl.ELEMENT_ARRAY_BUFFER. With everything
set up, you then call drawCube() at line 244, which uses gl.drawElements() at line 291 to
perform the draw operation.
drawTexCube(), defined at line 247, follows the same steps as drawSolidCube(). Line 253
is added to assign the buffer object for texture coordinates to the attribute variables, and
lines 257 and 258 are added to bind the texture object to the texture unit 0. The actual
drawing is performed in drawCube(), just like drawSolidCube().

Switching Shaders

391

Once you’ve mastered this basic technique, you can use it to switch between any number
of shader programs. This way you can use a variety of different drawing effects in a single
scene.

Use What You’ve Drawn as a Texture Image
One simple but powerful technique is to draw some 3D objects and then use the resulting image as a texture image for another 3D object. Essentially, if you can use the content
you’ve drawn as a texture image, you are able to generate images on-the-fly. This means
you do not need to download images from the network, and you can apply special effects
(such as motion blur and depth of field) before displaying the image. You can also use
this technique for shadowing, which will be explained in the next section. Here, you will
construct a sample program, FramebufferObject, which maps a rotating cube drawn with
WebGL to a rectangle as a texture image. Figure 10.17 shows a screen shot.

Figure 10.17

FramebufferObject

If you actually run the program, you can see a rotating cube with a texture image of a
summer sky pasted to the rectangle as its texture. Significantly, the image of the cube that
is pasted on the rectangle is not a movie prepared in advance but a rotating cube drawn
by WebGL in real time. This is quite powerful, so let’s take a look at what WebGL must do
to achieve this.

Framebuffer Object and Renderbuffer Object
By default, the WebGL system draws using a color buffer and, when using the hidden
surface removal function, a depth buffer. The final image is kept in the color buffer.
The framebuffer object is an alternative mechanism you can use instead of a color buffer
or a depth buffer (Figure 10.18). Unlike a color buffer, the content drawn in a framebuffer

392

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object is not directly displayed on the . Therefore, you can use it if you want to
perform different types of processing before displaying the drawn content. Or you can use
it as a texture image. Such a technique is often referred to as offscreen drawing.
per-vertex processing

per-fragment processing
Color Buffer

JavaScript
Vertex Shader
function main() {
var gl=getWebGL…
…
initShaders(…);
…
}

y

F ra gme nt
S ha de r

Framebuffer
Object

x

WebGL System
Figure 10.18

Framebuffer object

The framebuffer object has the structure shown in Figure 10.19 and supports substitutes
for the color buffer and the depth buffer. As you can see, drawing is not carried out in
the framebuffer itself, but in the drawing areas of the objects that the framebuffer points
to. These objects are attached to the framebuffer using its attachment function. A color
attachment specifies the destination for drawing to be a replacement for the color buffer.
A depth attachment and a stencil attachment specify the replacements for the depth
buffer and stencil buffer.

F r a m e b u ffe r
object

T e x tu re
o b je c t

Renderbuffer
object

C o lo r
a tta c h m e n t

Drawing
area

Drawing
area

Texture
object

Renderbuffer
object

Drawing
area

Drawing
area

D e p th
a tta c h m e n t
S te n c il
a tta c h m e n t

Figure 10.19

Framebuffer object, texture object, renderbuffer object

WebGL supports two types of objects that can be used to draw objects within: the texture
object that you saw in Chapter 5, and the renderbuffer object. With the texture object,
the content drawn into the texture object can be used as a texture image. The renderbuffer object is a more general-purpose drawing area, allowing a variety of data types to
be written.
Use What You’ve Drawn as a Texture Image

393

How to Implement Using a Drawn Object as a Texture
When you want to use the content drawn into a framebuffer object as a texture object,
you actually need to use the content drawn into the color buffer for the texture object.
Because you also want to remove the hidden surfaces for drawing, you will set up the
framebuffer object as shown in Figure 10.20.
F r a m e b u ffe r
object

T e x tu re
o b je c t

C o lo r
a tta c h m e n t

Drawing
area

D e p th
a tta c h m e n t
Renderbuffer
object

S te n c il
a tta c h m e n t

The size of each drawing area
must be identical.

Drawing
area

Figure 10.20

Configuration of framebuffer object when using drawn content as a texture

The following eight steps are needed for realizing this configuration. These processes are
similar to the process for the buffer object. Step 2 was explained in Chapter 5, so there are
essentially seven new processes:
1. Create a framebuffer object (gl.createFramebuffer()).
2. Create a texture object and set its size and parameters (gl.createTexture(),
gl.bindTexture(), gl.texImage2D(), gl.Parameteri()).
3. Create a renderbuffer object (gl.createRenderbuffer()).
4. Bind the renderbuffer object to the target and set its size (gl.bindRenderbuffer(),
gl.renderbufferStorage()).
5. Attach the texture object to the color attachment of the framebuffer object
(gl.bindFramebuffer(), gl.framebufferTexture2D()).
6. Attach the renderbuffer object to the depth attachment of the framebuffer object
(gl.framebufferRenderbuffer()).
7. Check whether the framebuffer object is configured correctly (gl.checkFramebufferStatus()).
8. Draw using the framebuffer object (gl.bindFramebuffer()).
Now let’s look at the sample program. The numbers in the sample program indicate the
code used to implement the steps.

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Sample Program (FramebufferObjectj.js)
Steps 1 to 7 of FramebufferObject.js are shown in Listing 10.13.
Listing 10.13
1
24
25
26
27
28
55
56
57
64
70
71
80
81
82
83
84
85

// FramebufferObject.js
...
// Size of offscreen
var OFFSCREEN_WIDTH = 256;
var OFFSCREEN_HEIGHT = 256;
function main() {
...
// Set vertex information
var cube = initVertexBuffersForCube(gl);
var plane = initVertexBuffersForPlane(gl);
...
var texture = initTextures(gl);
...
// Initialize framebuffer object (FBO)
var fbo = initFramebufferObject(gl);
...
var viewProjMatrix = new Matrix4();/ For color buffer
viewProjMatrix.setPerspective(30, canvas.width/canvas.height, 1.0, 100.0);
viewProjMatrix.lookAt(0.0, 0.0, 7.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0);
var viewProjMatrixFBO = new Matrix4(); // For FBO
viewProjMatrixFBO.setPerspective(30.0, OFFSCREEN_WIDTH/OFFSCREEN_HEIGHT,
➥1.0, 100.0);
viewProjMatrixFBO.lookAt(0.0, 2.0, 7.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0);
...
draw(gl, canvas, fbo, plane, cube, currentAngle, texture, viewProjMatrix,
➥viewProjMatrixFBO);
...

86
92

96
263
264
274
275
281
282

FramebufferObject.js (Processes for Steps 1 to 7)

}
...
function initFramebufferObject(gl) {
var framebuffer, texture, depthBuffer;
...
// Create a framebuffer object (FBO)
framebuffer = gl.createFramebuffer();
...
// Create a texture object and set its size and parameters
texture = gl.createTexture(); // Create a texture object
...

Use What You’ve Drawn as a Texture Image

<-(1)

<-(2)

395

287
288

gl.bindTexture(gl.TEXTURE_2D, texture);
gl.texImage2D(gl.TEXTURE_2D, 0, gl.RGBA, OFFSCREEN_WIDTH,
➥OFFSCREEN_HEIGHT, 0, gl.RGBA, gl.UNSIGNED_BYTE, null);
gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_MIN_FILTER, gl.LINEAR);
framebuffer.texture = texture;
// Store the texture object

289
290
291
292
293

// Create a renderbuffer object and set its size and parameters
depthBuffer = gl.createRenderbuffer();// Create a renderbuffer
<-(3)
...
gl.bindRenderbuffer(gl.RENDERBUFFER, depthBuffer);
<-(4)
gl.renderbufferStorage(gl.RENDERBUFFER, gl.DEPTH_COMPONENT16,
➥OFFSCREEN_WIDTH, OFFSCREEN_HEIGHT);

298
299
300
301
302
303

// Attach the texture and the renderbuffer object to the FBO
gl.bindFramebuffer(gl.FRAMEBUFFER, framebuffer);
gl.framebufferTexture2D(gl.FRAMEBUFFER, gl.COLOR_ATTACHMENT0,
➥gl.TEXTURE_2D, texture, 0);
gl.framebufferRenderbuffer(gl.FRAMEBUFFER, gl.DEPTH_ATTACHMENT,
➥gl.RENDERBUFFER, depthBuffer);

304
305
306
307
308
309
310
311
312
319
320

// Check whether FBO is configured correctly
var e = gl.checkFramebufferStatus(gl.FRAMEBUFFER);
if (e !== gl.FRAMEBUFFER_COMPLETE) {
console.log('Framebuffer object is incomplete: ' + e.toString());
return error();
}

<-(5)
<-(6)
<-(7)

...
return framebuffer;
}

The vertex shader and fragment shader are omitted because this sample program uses the
same shaders as TexturedQuad.js in Chapter 5, which pasted a texture image on a rectangle. The sample program in this section draws two objects: a cube and a rectangle. Just
as you did in ProgramObject.js in the previous section, you assign multiple buffer objects
needed for drawing each object as properties of an Object object. Then you store the
object to the variables cube and plane. You will use them for drawing by assigning each
buffer in the object to the attribute variable.
The key point of this program is the initialization of the framebuffer object by initFramebufferObject() at line 71. The initialized framebuffer object is stored in a variable
fbo and passed as the third argument of draw() at line 92. You’ll return to the function
draw() later. For now let’s examine initFramebufferObject(), at line 263, step by step.
This function performs steps 1 to 7. The view projection matrix for the framebuffer object
is prepared separately at line 84 because it is different from the one used for a color buffer.

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Create Frame Buffer Object (gl.createFramebuffer())
You must create a framebuffer object before you can use it. The sample program creates it
at line 275:
275 framebuffer = gl.createFramebuffer();

You will use gl.createFramebuffer() to create the framebuffer object.

gl.createFramebuffer()
Create a framebuffer object.
Parameters

None

Return value

non-null

The newly created framebuffer object.

null

Failed to create a framebuffer object.

Errors

None

You use gl.deleteFramebuffer() to delete the created framebuffer object.

gl.deleteFramebuffer(framebuffer)
Delete a framebuffer object.
Parameters

framebuffer

Return value

None

Errors

None

Specifies the framebuffer object to be deleted.

Once you have created the framebuffer object, you need to attach a texture object to the
color attachment and a renderbuffer object to the depth attachment in the framebuffer
object. Let’s start by creating the texture object for the color attachment.

Create Texture Object and Set Its Size and Parameters
You have already seen how to create a texture object and set up its parameters
(gl.TEXTURE_MIN_FILTER) in Chapter 5. You should note that its width and height are
OFFSCREEN_WIDTH and OFFSCREEN_HEIGHT, respectively. The size is smaller than that of the
 to make the drawing process faster.
282
287

texture = gl.createTexture(); // Create a texture object
...
gl.bindTexture(gl.TEXTURE_2D, texture);

Use What You’ve Drawn as a Texture Image

397

288
289
290

gl.texImage2D(gl.TEXTURE_2D, 0, gl.RGBA, OFFSCREEN_WIDTH, OFFSCREEN_HEIGHT, 0,
➥gl.RGBA, gl.UNSIGNED_BYTE, null);
gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_MIN_FILTER, gl.LINEAR);
framebuffer.texture = texture; // Store the texture object

The gl.texImage2D() at line 288 allocates a drawing area in a texture object. You can allocate a drawing area by specifying null to the last argument, which is used to specify an
Image object. You will use this texture object later, so store it in framebuffer.texture at
line 290.
That completes the preparation for a texture object that is attached to the color attachment. Next, you need to create a renderbuffer object for the depth buffer.

Create Renderbuffer Object (gl.createRenderbuffer())
Like texture buffers, you need to create a renderbuffer object before using it. The sample
program does this at line 293.
293

depthBuffer = gl.createRenderbuffer();

// Create a renderbuffer

You use gl.createRenderbuffer() to create the renderbuffer object.

gl.createRenderbuffer()
Create a renderbuffer object.
Parameters

None

Return value

Non-null

The newly created renderbuffer object.

Null

Failed to create a renderbuffer object.

Errors

None

You use gl.deleteRenderbuffer() to delete the created renderbuffer object.

gl.deleteRenderbuffer(renderbuffer)
Delete a renderbuffer object.
Parameters

renderbuffer

Return value

None

Errors

None

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Specifies the renderbuffer object to be deleted.

Advanced Techniques

The created renderbuffer object is used as a depth buffer here, so you store it in a variable
named depthBuffer.

Bind Renderbuffer Object to Target and Set Size
(gl.bindRenderbuffer(), gl.renderbufferStorage())
When using the created renderbuffer object, you need to bind the renderbuffer object to a
target and perform the operation on that target.
298
299

gl.bindRenderbuffer(gl.RENDERBUFFER, depthBuffer);
gl.renderbufferStorage(gl.RENDERBUFFER, gl.DEPTH_COMPONENT16,
➥OFFSCREEN_WIDTH, OFFSCREEN_HEIGHT);

The renderbuffer object is bound to a target with gl.bindRenderbuffer().

gl.bindRenderbuffer(target, renderbuffer)
Bind the renderbuffer object specified by renderbuffer to target. If null is specified as
renderbuffer, the renderbuffer is unbound from the target.
Parameters

target

Must be gl.RENDERBUFFER.

renderbuffer

Specifies the renderbuffer object.

Return value

None

Errors

INVALID_ENUM

target is not gl.RENDERBUFFER

When the binding is complete, you can set the format, width, and height of the renderbuffer object by using gl.renderbufferStorage(). You must set the same width and
height as the texture object that is used as the color attachment.

gl.renderbufferStorage(target, internalformat, width, height)
Create and initialize a renderbuffer object’s data store.
Parameters

target

Must be gl.RENDERBUFFER.

internalformat

Specifies the format of the renderbuffer.

gl.DEPTH_
COMPONENT16

The renderbuffer is used as a depth buffer.

gl.STENCIL_

The renderbuffer is used as a stencil buffer.

INDEX8

Use What You’ve Drawn as a Texture Image

399

gl.RGBA4
gl.RGB5_A1
gl.RGB565

width, height

The renderbuffer is used as a color buffer. gl.RGBA4
(each RGBA component has 4, 4, 4, and 4 bits, respectively), gl.RGB5_A1 (each RGB component has 5 bits,
and A has 1 bit), gl.RGB565 (each RGB component has
5, 6, and 5 bits, respectively)
Specifies the width and height of the renderbuffer in
pixels.

Return value None
Errors

INVALID_ENUM

Target is not gl.RENDERBUFFER or internalformat is none
of the preceding values.

INVALID_OPERATION

No renderbuffer is bound to target.

The preparations of the texture object and renderbuffer object of the framebuffer object
are now complete. At this stage, you can use the object for offscreen drawing.

Set Texture Object to Framebuffer Object (gl.bindFramebuffer(),
gl.framebufferTexture2D())
You use a framebuffer object in the same way you use a renderbuffer object: You need to
bind it to a target and operate on the target, not the framebuffer object itself.
302
303

gl.bindFramebuffer(gl.FRAMEBUFFER, framebuffer); // Bind to target
gl.framebufferTexture2D(gl.FRAMEBUFFER, gl.COLOR_ATTACHMENT0, gl.TEXTURE_2D,
➥texture, 0);

A framebuffer object is bound to a target with gl.bindFramebuffer().

gl.bindFramebuffer(target, framebuffer)
Bind a framebuffer object to a target. If framebuffer is null, the binding is broken.
Parameters

target

Must be gl.FRAMEBUFFER.

framebuffer

Specify the framebuffer object.

Return value

None

Errors

INVALID_ENUM

target is not gl.FRAMEBUFFER

Once the framebuffer object is bound to target, you can use the target to write a texture
object to the framebuffer object. In this sample, you will use the texture object instead of
a color buffer so you attach the texture object to the color attachment of the framebuffer.

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You can assign the texture object to the framebuffer object with gl.
framebufferTexture2D().

gl.framebufferTexture2D(target, attachment, textarget, texture,
level)
Attach a texture object specified by texture to the framebuffer object bound by target.
Parameters

target

Must be gl.FRAMEBUFFER.

attachment

Specifies the attachment point of the framebuffer.

gl.COLOR_ATTACHMENT0

texture is used as a color buffer

gl.DEPTH_ATTACHMENT

texture is used as a depth buffer

textarget

Specifies the first argument of gl.texImage2D()
(gl.TEXTURE_2D or gl.CUBE_MAP_TEXTURE).

texture

Specifies a texture object to attach to the framebuffer attachment point.

level

Specifies 0 (if you use a MIPMAP in texture, you
should specify its level).

Return value

None

Errors

INVALID_ENUM

target is not gl.FRAMEBUFFER. attachment
or textarget is none of the preceding values.

INVALID_VALUE

level is not valid.

INVALID_OPERATION

No framebuffer object is bound to target.

The 0 in the gl.COLOR_ATTACHMENT0 used for the attachment parameter is because a framebuffer object in OpenGL, the basis of WebGL, can hold multiple color attachments
(gl.COLOR_ATTACHMENT0, gl.COLOR_ATTACHMENT1, gl.COLOR_ATTACHMENT2...). However,
WebGL can use just one of them.
Once the color attachment has been attached to the framebuffer object, you need to
assign a renderbuffer object as a depth attachment. This follows a similar process.

Set Renderbuffer Object to Framebuffer Object
(gl.framebufferRenderbuffer())
You will use gl.framebufferRenderbuffer() to attach a renderbuffer object to a framebuffer object. You need a depth buffer because this sample program will remove hidden
surfaces. So the depth attachment needs to be attached.

Use What You’ve Drawn as a Texture Image

401

304
gl.framebufferRenderbuffer(gl.FRAMEBUFFER, gl.DEPTH_ATTACHMENT,
gl.RENDERBUFFER, depthBuffer);

gl.framebufferRenderbuffer(target, attachment, renderbuffertarget,
renderbuffer)
Attach a renderbuffer object specified by renderbuffer to the framebuffer object bound by
target.
Parameters

target

Must be gl.FRAMEBUFFER.

attachment

Specifies the attachment point of the framebuffer.

gl.COLOR_ATTACHMENT0

renderbuffer is used as a color buffer.

gl.DEPTH_ATTACHMENT

renderbuffer is used as a depth buffer.

gl.STENCIL_ATTACHMENT

renderbuffer is used as a stencil buffer.

renderbuffertarget

Must be gl.RENDERBUFFER.

renderbuffer

Specifies a renderbuffer object to attach to the
framebuffer attachment point

Return value None
Errors

INVALID_ENUM

target is not a gl.FRAMEBUFFER. attachment is
none of the above values. renderbuffertarget is
not gl.RENDERBUFFER.

Now that you’ve completed the preparation of the color attachment and depth attachment to the framebuffer object, you are ready to draw. But before that, let’s check that the
configuration of the framebuffer object is correct.

Check Configuration of Framebuffer Object
(gl.checkFramebufferStatus())
Obviously, when you use a framebuffer that is not correctly configured, an error occurs. As
you have seen in the past few sections, preparing a texture object and renderbuffer object
that are needed to configure the framebuffer object is a complex process that sometimes
generates mistakes. You can check whether the created framebuffer object is configured
correctly and is available with gl.checkFramebufferStatus().
307
308
309
310
311
402

var e = gl.checkFramebufferStatus(gl.FRAMEBUFFER);
<- (7)
if (gl.FRAMEBUFFER_COMPLETE !== e) {
console.log('Frame buffer object is incomplete:' + e.toString());
return error();
}

CHAPTER 10

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The following shows the specification of gl.checkFramebufferStatus().

gl.checkFramebufferStatus(target)
Check the completeness status of a framebuffer bound to target.
Parameters

target

Must be gl.FRAMEBUFFER.

Return value

0

Target is not gl.FRAMEBUFFER.

Others

Errors

gl.FRAMEBUFFER_COMPLETE

The framebuffer object is configured
correctly.

gl.FRAMEBUFFER_INCOMPLETE_
ATTACHMENT

One of the framebuffer attachment points
is incomplete. (The attachment is not sufficient. The texture object or the renderbuffer object is invalid.)

gl.FRAMEBUFFER_INCOMPLETE_
DIMENSIONS

The width or height of the texture object
or renderbuffer object of the attachment is
different.

gl.FRAMEBUFFER_INCOMPLETE_
MISSING_ATTACHMENT

The framebuffer does not have at least
one valid attachment.

INVALID_ENUM

target is not gl.FRAMEBUFFER.

That completes the preparation of the framebuffer object. Let’s now take a look at the
draw() function.

Draw Using the Framebuffer Object
Listing 10.14 shows draw(). It switches the drawing destination to fbo (the framebuffer)
and draws a cube in the texture object. Then drawTexturedPlane() uses the texture object
to draw a rectangle to the color buffer.
Listing 10.14
321
322
323
324
325
326

FramebufferObject.js (Process of (8))

function draw(gl, canvas, fbo, plane, cube, angle, texture, viewProjMatrix,
➥viewProjMatrixFBO) {
gl.bindFramebuffer(gl.FRAMEBUFFER, fbo);
<-(8)
gl.viewport(0, 0, OFFSCREEN_WIDTH, OFFSCREEN_HEIGHT); // For FBO
gl.clearColor(0.2, 0.2, 0.4, 1.0); // Color is slightly changed
gl.clear(gl.COLOR_BUFFER_BIT | gl.DEPTH_BUFFER_BIT); // Clear FBO

Use What You’ve Drawn as a Texture Image

403

327
328
329
330
331
332
333
334
335
336
337

// Draw the cube
drawTexturedCube(gl, gl.program, cube, angle, texture, viewProjMatrixFBO);
// Change the drawing destination to color buffer
gl.bindFramebuffer(gl.FRAMEBUFFER, null);
// Set the size of view port back to that of 
gl.viewport(0, 0, canvas.width, canvas.height);
gl.clearColor(0.0, 0.0, 0.0, 1.0);
gl.clear(gl.COLOR_BUFFER_BIT | gl.DEPTH_BUFFER_BIT);
// Draw the plane
drawTexturedPlane(gl, gl.program, plane, angle, fbo.texture, viewProjMatrix);
}

Line 322 switches the drawing destination to the framebuffer object using gl.bindFramebuffer(). As a result, draw operations using gl.drawArrays() or gl.drawElements() are
performed for the framebuffer object. Line 332 uses gl.viewport() to specify the draw
area in the buffer (an offscreen area).

gl.viewport(x, y, width, height)
Set the viewport where gl.drawArrays() or gl.drawElements() draws. In WebGL, x and
y are specified in the  coordinate system.
Parameters

x, y

Specify the lower-left corner of the viewport rectangle (in
pixels).

width, height

Specify the width and height of the viewport (in pixels).

Return value

None

Errors

None

Line 326 clears the texture image and the depth buffer bound to the framebuffer object.
When a cube is drawn at line 328, it is drawn in the texture image. To make it easier to
see the result, the clear color at line 325 is changed to a purplish blue from black. The
result of this is that the cube has been drawn into the texture buffer and is now available
for use as a texture image. The next step is to draw a rectangle (plane) using this texture
image. In this case, because you want to draw in the color buffer, you need to set the
drawing destination back to the color buffer. This is done at line 330 by specifying null
for the second argument of gl.bindFramebuffer() (that is, cancelling the binding). Then
line 336 draws the plane. You should note that fbo.texture is passed as the texture argument and used to map the drawn content to the rectangle. You will notice that in this
sample program, the texture image is mapped onto the back side of the rectangle. This is
because WebGL, by default, draws both sides of a polygon. You can eliminate the back
face drawing by enabling the culling function using gl.enable(gl.CULL_FACE), which
increases the drawing speed (ideally making it twice as fast).

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Display Shadows
Chapter 8 explained shading, which is one of the phenomena when light hits an object.
We briefly mentioned shadowing, another phenomena, but didn’t explain how to implement it. Let’s take a look at that now. There are several methods to realize shadowing,
but we will explain a method that uses a shadow map (depth map). This method is quite
expressive and used in a variety of computer graphics situations and even in special effects
in movies.

How to Implement Shadows
The shadow map method is based on the idea that the sun cannot see the shadow of
objects. Essentially, it works by considering the viewer’s eye point to be at the same position as the light source and determining what can be seen from that point. All the objects
you can see would appear to be in the light. Anything behind those objects would be in
shadow. With this method, you can use the distance to the objects (in fact, you will use
the z value, which is the depth value) from the light source to judge whether the objects
are visible. As you can see in Figure 10.21, where there are two points on the same line,
P1 and P2, P2 is in the shadow because the distance from the light source to P2 is longer
than P1.

y

p1

p2

T he ar ea dar k ened by shadow

z
Figure 10.21

Theory of shadow map

You need two pairs of shaders for this process: (1) a pair of shaders that calculate the
distance from the light source to the objects, and (2) a pair of shaders that draws the
shadow using the calculated distance. Then you need a method to pass the distance data
from the light source calculated in the first pair of shaders to the second pair of shaders.
You can use a texture image for this purpose. This texture image is called the shadow
map, so this method is called shadow mapping. The shadow mapping technique consists
of the following two processes:
1. Move the eye point to the position of the light source and draw objects from there.
Because the fragments drawn from the position are hit by the light, you write the
distances from the light source to each fragment in the texture image (shadow map).
2. Move the eye point back to the position from which you want to view the objects
and draw them from there. Compare the distance from the light source to the fragments drawn in this step and the distance recorded in the shadow map from step
Display Shadows

405

1. If the former distance is greater, you can draw the fragment as in shadow (in the
darker color).
You will use the framebuffer object in step 1 to save the distance in the texture image.
Therefore, the configurations of the framebuffer object used here is the same as that of
FramebufferObject.js in Figure 10.20. You also need to switch pairs of shaders between
steps 1 and 2 using the technique you learned in the section “Switching Shaders,” earlier
in this chapter. Now let’s take a look at the sample program Shadow. Figure 10.22 shows
a screen shot where you can see a shadow of the red triangle cast onto the slanted white
rectangle.

Figure 10.22

Shadow

Sample Program (Shadow.js)
The key aspects of shadowing take place in the shaders, which are shown in Listing 10.15.
Listing 10.15
1
2
3
6
7
8
9
10
11

406

Shadow.js (Shader part)

// Shadow.js
// Vertex shader program to generate a shadow map
var SHADOW_VSHADER_SOURCE =
...
'void main() {\n' +
' gl_Position = u_MvpMatrix * a_Position;\n' +
'}\n';
// Fragment shader program for creating a shadow map
var SHADOW_FSHADER_SOURCE =

CHAPTER 10

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15
16
17
18
19
20
23
24
25
26
27
28
29
30
31
32
33
34
38
39
40
41
42
43
44
45
46
47

...
'void main() {\n' +
' gl_FragColor = vec4(gl_FragCoord.z, 0.0, 0.0, 0.0);\n' +
'}\n';

<-(1)

// Vertex shader program for regular drawing
var VSHADER_SOURCE =
...
'uniform mat4 u_MvpMatrix;\n' +
'uniform mat4 u_MvpMatrixFromLight;\n' +
'varying vec4 v_PositionFromLight;\n' +
'varying vec4 v_Color;\n' +
'void main() {\n' +
' gl_Position = u_MvpMatrix * a_Position;\n' +
' v_PositionFromLight = u_MvpMatrixFromLight * a_Position;\n' +
' v_Color = a_Color;\n' +
'}\n';
// Fragment shader program for regular drawing
var FSHADER_SOURCE =
...
'uniform sampler2D u_ShadowMap;\n' +
'varying vec4 v_PositionFromLight;\n' +
'varying vec4 v_Color;\n' +
'void main() {\n' +
' vec3 shadowCoord =(v_PositionFromLight.xyz/v_PositionFromLight.w)
➥/ 2.0 + 0.5;\n' +
' vec4 rgbaDepth = texture2D(u_ShadowMap, shadowCoord.xy);\n' +
' float depth = rgbaDepth.r;\n' + // Retrieve the z value from R
' float visibility = (shadowCoord.z > depth + 0.005) ? 0.7:1.0;\n'+
<-(2)
' gl_FragColor = vec4(v_Color.rgb * visibility, v_Color.a);\n' +
'}\n';

Step 1 is performed in the shader responsible for the shadow map, defined from lines 3 to
17. You just switch the drawing destination to the framebuffer object, pass a model view
projection matrix in which an eye point is located at a light source to u_MvpMatrix, and
draw the objects. This results in the distance from the light source to the fragments being
written into the texture map (shadow map) attached to the framebuffer object. The vertex
shader at line 7 just multiplies the model view projection matrix by the vertex coordinates
to calculate this distance. The fragment shader is more complex and needs to calculate the
distance from the light source to the drawn fragments. For this purpose, you can utilize
the built-in variable gl_FragCoord of the fragment shader used in Chapter 5.
gl_FragCoord is a vec4 type built-in variable that contains the coordinates of each fragment. gl_FragCoord.x and gl_FragCoord.y represents the position of the fragment on the

Display Shadows

407

screen, and gl_FragCoord.z contains the normalized z value in the range of [0, 1]. This is
calculated using (gl_Position.z/gl.Position.w)/2.0+0.5. (See Section 2.12 of OpenGL ES
2.0 specification for further details.) gl_FragCoord.z is specified in the range of 0.0 to 1.0,
with 0.0 representing the fragments on the near clipping plane and 1.0 representing those
on the far clipping plane. This value is written into the R (red) component value (any
component could be used) in the shadow map at line 16.
16

'

gl_FragColor = vec4(gl_FragCoord.z, 0.0, 0.0, 0.0);\n'

+

<-(1)

Subsequently, the z value for each fragment drawn from the eye point placed at the light
source is written into the shadow map. This shadow map is passed to u_ShadowMap at
line 38.
For step 2, you need to draw the objects again after resetting the drawing destination
to the color buffer and moving the eye point to its original position. After drawing the
objects, you decide a fragment color by comparing the z value of the fragment with
that stored in the shadow map. This is done in the normal shaders from lines 20 to 47.
u_MvpMatrix is the model view projection matrix where the eye point is placed at the original position and uMvpMatrixFromLight, which was used to create the shadow map, is the
model view projection matrix where the eye point is moved to the light source. The main
task of the vertex shader defined at line 20 is calculating the coordinates of each fragment
from the light source and passing them to the fragment shader (line 29) to obtain the z
value of each fragment from the light source.
The fragment shader uses the coordinates to calculate the z value (line 42). As mentioned,
the shadow map contains the value of (gl_Position.z/gl.Position.w)/2.0+0.5. So
you could simply calculate the z value to compare with the value in the shadow map
by (v_PositionFromLight.z/v_PositionFromLight.w)/2.0+0.5. However, because you
need to get the texel value from the shadow map, line 42 performs the following extra
calculation using the same operation. To compare to the value in the shadow map, you
need to get the texel value from the shadow map whose texture coordinates correspond
to the coordinates (v_PositionFromLight.x, v_PositionFromLight.y). As you know,
v_PositionFromLight.x and v_PositionFromLight.y are the x and y coordinates in the
WebGL coordinate system (see Figure 2.18 in Chapter 2), and they range from –1.0 to 1.0.
On the other hand, the texture coordinates s and t in the shadow map range from 0.0 to
1.0 (see Figure 5.20 in Chapter 5). So, you need to convert the x and y coordinates to the s
and t coordinates. You can also do this with the same expression to calculate the z value.
That is:
The texture coordinate s is (v_PositionFromLight.x/v_PositionFromLight.w)/2.0 + 0.5.
The texture coordinate t is (v_PositionFromLight.y/v_PositionFromLight.w)/2.0 + 0.5.
See also Section 2.12 of the OpenGL ES 2.0 specification5 for further details about this calculation. These are carried out using the same type of calculation and can be achieved in
one line, as shown at line 42:
5

www.khronos.org/registry/gles/specs/2.0/es_full_spec_2.0.25.pdf

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42

'

43
44

'
'

vec3 shadowCoord =(v_PositionFromLight.xyz/v_PositionFromLight.w)
➥/ 2.0 + 0.5;\n' +
vec4 rgbaDepth = texture2D(u_ShadowMap, shadowCoord.xy);\n' +
float depth = rgbaDepth.r;\n' + // Retrieve the z value from R

You retrieve the value from the shadow map at lines 43 and 44. Only the R value is
retrieved using rgbaDepth.r at line 44 because you wrote it into the R component at line
16. Line 45 checks whether that fragment is in the shadow. When the position of the
fragment is determined to be greater than the depth (that is, shadowCoord.z > depth), a
value of 0.7 is stored in visibility. The visibility is used at line 46 to draw the shadow
with a darker color:
45
46

'
'

float visibility = (shadowCoord.z > depth + 0.005) ? 0.7:1.0;\n'+
gl_FragColor = vec4(v_Color.rgb * visibility, v_Color.a);\n' +

Line 45 adds a small offset of 0.005 to the depth value. To understand why this is needed,
try running the sample program without this number. You will see a striped pattern as
shown in Figure 10.23, referred to as the Mach band.

Figure 10.23

Striped pattern

The value of 0.005 is added to suppress the stripe pattern. The stripe pattern occurs
because of the precision of the numbers you can store in the RGBA components. It’s a
little complex, but it’s worth understanding because this problem occurs elsewhere in 3D
graphics. The z value of the shadow map is stored in the R component of RGBA in the
texture map, which is an 8-bit number. This means that the precision of R is lower than
its comparison target (shadowCoord.z), which is of type float. For example, let the z value
simply be 0.1234567. If you represent the value using 8 bits, in other words using 256

Display Shadows

409

possibilities, you can represent the value in a precision of 1/256 (=0.0390625). So you can
represent 0.1234567 as follows:
0.1234567 / (1 / 256) = 31.6049152

Numbers below the decimal point cannot be used in 8 bits, so only 31 can be stored
in 8 bits. When you divide 31 by 256, you obtain 0.12109375 which, as you can see, is
smaller than the original value (0.1234567). This means that even if the fragment is at
the same position, its z value stored in the shadow map becomes smaller than its z value
in shadowCoord.z. As a result, the z value in shadowCoord.z becomes larger than that in
the shadow map according to the position of the fragment resulting in the stripe patterns.
Because this happens because the precision of the R value is 1/256 (=0.00390625), by
adding a small offset, such as 0.005, to the R value, you can stop the stripe pattern from
appearing. Note that any offset greater than 1/256 will work; 0.005 was chosen because it
is 1/256 plus a small margin.
Next, let’s look at the JavaScript program that passes the data to the shader (see Listing
10.16) with a focus on the type of transformation matrices passed. To draw a shadow
clearly, the size of a texture map for the offscreen rendering defined at line 49 is larger
than that of the .
Listing 10.16
49
50
51
52
63
64

72
73
85
86
87
93
94
99
100
106

410

Shadow.js (JavaScript Part)

var OFFSCREEN_WIDTH = 1024, OFFSCREEN_HEIGHT = 1024;
var LIGHT_X = 0, LIGHT_Y = 7, LIGHT_Z = 2;
function main() {
...
// Initialize shaders for generating a shadow map
var shadowProgram = createProgram(gl, SHADOW_VSHADER_SOURCE,
➥SHADOW_FSHADER_SOURCE);
...
// Initialize shaders for regular drawing
var normalProgram = createProgram(gl, VSHADER_SOURCE, FSHADER_SOURCE);
...
// Set vertex information
var triangle = initVertexBuffersForTriangle(gl);
var plane = initVertexBuffersForPlane(gl);
...
// Initialize a framebuffer object (FBO)
var fbo = initFramebufferObject(gl);
...
gl.activeTexture(gl.TEXTURE0); // Set a texture object to the texture unit
gl.bindTexture(gl.TEXTURE_2D, fbo.texture);
...
var viewProjMatrixFromLight = new Matrix4(); // For the shadow map

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107

viewProjMatrixFromLight.setPerspective(70.0,
➥OFFSCREEN_WIDTH/OFFSCREEN_HEIGHT, 1.0, 100.0);
viewProjMatrixFromLight.lookAt(LIGHT_X, LIGHT_Y, LIGHT_Z, 0.0, 0.0, 0.0, 0.0,
➥1.0, 0.0);

108
109
110
111
112
113
114
115
116
117
118
119
120

var viewProjMatrix = new Matrix4(); // For regular drawing
viewProjMatrix.setPerspective(45, canvas.width/canvas.height, 1.0, 100.0);
viewProjMatrix.lookAt(0.0, 7.0, 9.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0);
var currentAngle = 0.0; // Current rotation angle [degrees]
var mvpMatrixFromLight_t = new Matrix4(); // For triangle
var mvpMatrixFromLight_p = new Matrix4(); // For plane
var tick = function() {
currentAngle = animate(currentAngle);
// Change the drawing destination to FBO
gl.bindFramebuffer(gl.FRAMEBUFFER, fbo);
...
gl.useProgram(shadowProgram); // For generating a shadow map
// Draw the triangle and the plane (for generating a shadow map)
drawTriangle(gl, shadowProgram, triangle, currentAngle,
➥viewProjMatrixFromLight);
mvpMatrixFromLight_t.set(g_mvpMatrix); // Used later
drawPlane(gl, shadowProgram, plane, viewProjMatrixFromLight);
mvpMatrixFromLight_p.set(g_mvpMatrix); // Used later
// Change the drawing destination to color buffer
gl.bindFramebuffer(gl.FRAMEBUFFER, null);
...
gl.useProgram(normalProgram); // For regular drawing
gl.uniform1i(normalProgram.u_ShadowMap, 0); // Pass gl.TEXTURE0
// Draw the triangle and plane (for regular drawing)
gl.uniformMatrix4fv(normalProgram.u_MvpMatrixFromLight, false,
➥mvpMatrixFromLight_t.elements);
drawTriangle(gl, normalProgram, triangle, currentAngle, viewProjMatrix);
gl.uniformMatrix4fv(normalProgram.u_MvpMatrixFromLight, false,
➥mvpMatrixFromLight_p.elements);
drawPlane(gl, normalProgram, plane, viewProjMatrix);

124
125
126
127
128
129
130
131
135
136
137
138
139
140
141
142
143
144
145
146

window.requestAnimationFrame(tick, canvas);
};
tick();
}

Let’s look at the main() function from line 52 in the JavaScript program. Line 64
initializes the shaders for generating the shadow map. Line 73 initializes the shaders
for normal drawing. Lines 86 and 87, which set up the vertex information and
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411

initFramebufferObject() at line 94, are the same as the FramebufferObject.js. Line 94
prepares a framebuffer object, which contains the texture object for a shadow map. Lines
99 and 100 enable texture unit 0 and bind it to the target. This texture unit is passed to
u_ShadowMap in the shaders for normal drawing.

Lines 106 to 108 prepare a view projection matrix to generate a shadow map. The key
point is that the first three arguments (that is, the position of an eye point) at line 108 are
specified as the position of the light source. Lines 110 to 112 prepare the view projection
matrix from the eye point where you want to view the scene.
Finally, you draw the triangle and plane using all the preceding information. First you
generate the shadow map, so you switch the drawing destination to the framebuffer
object at line 120. You draw the objects by using the shaders for generating a shadow map
(shadowProgram) at lines 126 and 128. You should note that lines 127 and 129 save the
model view projection matrices from the light source. Then the shadow map is generated,
and you use it to draw shadows with the code from line 135. Line 136 passes the map to
the fragment shader. Lines 138 and 140 pass the model view projection matrices saved at
line 127 and 129, respectively, to u_MvpMatrixFromLight.

Increasing Precision
Although you’ve successfully calculated the shadow and drawn the scene with the shadow
included, the example code is only able to handle situations in which the light source is
close to the object. To see this, let’s change the y coordinate of the light source position
to 40:
50

var LIGHT_X = 0, LIGHT_Y = 40, LIGHT_Z = 2;

If you run the modified sample program, you can see that the shadow is not displayed—as
in the left side of Figure 10.24. Obviously, you want the shadow to be displayed correctly,
as in the figure on the right.
The reason the shadow is no longer displayed when the distance from the light source to
the object is increased is that the value of gl_FragCoord.z could not be stored in the R
component of the texture map because it has only an 8-bit precision. A simple solution to
this problem is to use not just the R component but the B, G, and A components. In other
words, you save the value separately in 4 bytes. There is a routine procedure to do this, so
let’s see the sample program. Only the fragment shader is changed.

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Figure 10.24

The shadow is not displayed

Sample Program (Shadow_highp.js)
Listing 10.17 shows the fragment shader of Shadow_highp.js. You can see that the
processing to handle the z value is more complex than that in Shadow.js.
Listing 10.17
1
10
11
15
16
17
18
19
20
21
37
38
45
46
47

Shadow_highp.js

// Shadow_highp.js
...
// Fragment shader program for creating a shadow map
var SHADOW_FSHADER_SOURCE =
...
'void main() {\n' +
' const vec4 bitShift = vec4(1.0, 256.0, 256.0 * 256.0, 256.0 * 256.0 *
➥256.0);\n' +
' const vec4 bitMask = vec4(1.0/256.0, 1.0/256.0, 1.0/256.0, 0.0);\n' +
' vec4 rgbaDepth = fract(gl_FragCoord.z * bitShift);\n' +
' rgbaDepth -= rgbaDepth.gbaa * bitMask;\n' +
' gl_FragColor = rgbaDepth;\n' +
'}\n';
...
// Fragment shader program for regular drawing
var FSHADER_SOURCE =
...
// Recalculate the z value from the rgba
'float unpackDepth(const in vec4 rgbaDepth) {\n' +
' const vec4 bitShift = vec4(1.0, 1.0/256.0, 1.0/(256.0 * 256.0),
➥1.0/(256.0 * 256.0 * 256.0));\n' +

Display Shadows

413

48
49
50
51
52
53
54
55
56
57

' float depth = dot(rgbaDepth, bitShift);\n' +
' return depth;\n' +
'}\n' +
'void main() {\n' +
' vec3 shadowCoord = (v_PositionFromLight.xyz /
➥v_PositionFromLight.w)/2.0 + 0.5;\n' +
' vec4 rgbaDepth = texture2D(u_ShadowMap, shadowCoord.xy);\n' +
' float depth = unpackDepth(rgbaDepth);\n' + // Recalculate the z
' float visibility = (shadowCoord.z > depth + 0.0015)? 0.7:1.0;\n'+
' gl_FragColor = vec4(v_Color.rgb * visibility, v_Color.a);\n' +
'}\n';

The code that splits gl_FragCoord.z into 4 bytes (RGBA) is from lines 16 to 19. Because 1
byte can represent up to 1/256, you can store the value greater than 1/256 in R, the value
less than 1/256 and greater than 1/(256*256) in G, the value less than 1/(256*256) and
greater than 1/(256*256*256) in B, and the rest of value in A. Line 18 calculates each value
and stores it in the RGBA components, respectively. It can be written in one line using a
vec4 data type. The function fract() is a built-in one that discards numbers below the
decimal point for the value specified as its argument. Each value in vec4, calculated at
line 18, has more precision than 1 byte, so line 19 discards the value that does not fit in 1
byte. By substituting this result to gl_FragColor at line 20, you can save the z value using
all four components of the RGBA type and achieve higher precision.
unpackDepth() at line 54 reads out the z value from the RGBA. This function is defined at

line 46. Line 48 performs the following calculation to convert the RGBA value to the original z value. As you can see, the calculation is the same as the inner product, so you use
dot() at line 48.
depth = rgbDepth.r × 1.0 +

rgbaDepth.g
rgbaDepth.b
rgbaDepth.a
+
+
256.0
( 256.0 × 256.0 ) ( 256.0 × 256.0 × 256.0 )

Now you have retrieved the distance (z value) successfully, so you just have to draw the
shadow by comparing the distance with shadowCoord.z at line 55. In this case, 0.0015
is used as the value for adjusting the error (the stripe pattern), instead of 0.005. This is
because the precision of the z value stored in the shadow map is a float type of medium
precision (that is, its precision is 2–10 = 0.000976563, as shown in Table 6.15 in Chapter 6).
So you add a little margin to it and chose 0.0015 as the value. After that, the shadow can
be drawn correctly.

Load and Display 3D Models
In the previous chapters, you drew 3D objects by specifying their vertex coordinates
and color information by hand and stored them in arrays of type Float32Array in
the JavaScript program. However, as mentioned earlier in the book, in most cases you
will actually read the vertex coordinates and color information from 3D model files
constructed by a 3D modeling tool.
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In this section, you construct a sample program that reads a 3D model constructed using
a 3D modeling tool. For this example, we use the Blender6 modeling tool, which is a
popular tool with a free version available. Blender is able to export 3D model files using
the well-known OBJ format, which is text based and easy to read, understand, and parse.
OBJ is a geometry definition file format originally developed by Wavefront Technologies.
This file format is open and has been adopted by other 3D graphics vendors. Although
this means it is reasonably well known and used, it also means that there are a number
of variations in the format. To simplify the example code, we have made a number of
assumptions, such as not using textures. However, the example gives you a good understanding of how to read model data into your programs and provides a basis for you to
begin experimentation. The approach taken in the example code is designed to be reasonably generic and can be used for other text-based formats.
Start Blender and create a cube like that shown in Figure 10.25. The color of one face of
this cube is orange, and the other faces are red. Then export the model to a file named
cube.obj. (You can find an example of it in the resources directory with the sample
programs.) Let’s take a look at cube.obj, which, because it is a text file, can be opened
with a simple text editor.

Figure 10.25

6.

Blender, 3D modeling tool

www.blender.org/

Load and Display 3D Models

415

Figure 10.26 shows the contents of cube.obj. Line numbers have been added to help with
the explanation and would not normally be in the file.
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0

# B le n d e r v 2 . 6 0 ( s u b 0 ) O B J F ile : ''
# w w w .b le n d e r .o r g
m tllib c u b e .m tl
o C ube
v 1 .0 0 0 0 0 0 -1 .0 0 0 0 0 0 -1 .0 0 0 0 0 0
v 1 . 0 0 0 0 0 0 - 1 . 0 0 0 0 0 0 1 .0 0 0 0 0 0
v -1 .0 0 0 0 0 0 -1 .0 0 0 0 0 0 1 .0 0 0 0 0 0
v -1 .0 0 0 0 0 0 -1 .0 0 0 0 0 0 -1 .0 0 0 0 0 0
v 1 .0 0 0 0 0 0 1 . 0 0 0 0 0 0 - 1 . 0 0 0 0 0 0
v 1 .0 0 0 0 0 0 1 .0 0 0 0 0 0 1 .0 0 0 0 0 1
v -1 .0 0 0 0 0 0 1 .0 0 0 0 0 0 1 .0 0 0 0 0 0
v -1 .0 0 0 0 0 0 1 .0 0 0 0 0 0 -1 .0 0 0 0 0 0
u s e m tl M a te r ia l
f1234
f5876
f2673
f3784
f5148
u s e m t l M a t e r ia l. 0 0 1
f1562

Figure 10.26

cube.obj

Once the model file has been created by the modeling tool, your program needs to read
the data and store it in the same type of data structures that you’ve used before. The
following steps are required:
1. Prepare the array (vertices) of type Float32Array and read the vertex coordinates of
the model from the file into the array.
2. Prepare the array (colors) of type Float32Array and read the colors of the model
from the file into the array.
3. Prepare the array (normals) of type Float32Array and read the normals of the model
form the file into the array.
4. Prepare the array (indices) of type Uint16Array (or Uint8Array) and read the indices
of the vertices that specify the triangles that make up the model from the file into
the array.
5. Write the data read during steps 1 through 4 into the buffer object and then draw
the model using gl.drawElements().
So in this case, you read the data described in cube.obj (shown in Figure 10.26) in the
appropriate arrays and then draw the model in step 5. Reading data from the file requires
understanding the format of the file cube.obj (referred to as the OBJ file).

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The OBJ File Format
An OBJ file is made up of several sections,7 including vertex positions, face definitions,
and material definitions. There may be multiple vertices, normals, and faces within their
sections:

• Lines beginning with a hash character (#) are comments. Lines 1 and 2 in Figure
10.26 are comments generated by Blender describing its version number and origin.
The remaining lines define the 3D model.

• Line 3 references an external materials file. The OBJ format maintains the material
information of the model in an external material file called an MTL file.
mtllib 

specifies that the materials file is cube.mtl.

• Line 4 specifies the named object in the following format: