Curves And Surfaces For CAGD: A Practical Guide Curves+and+Surfaces+for+CAGD +A+Practical+Guide,+Fifth+Editi

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- Copyright Page
Contents
- Preface
Chapter 1. P. Bézier: How a Simple System Was Born
Chapter 2. Introductory Material
- 2.1 Points and Vectors
- 2.2 Affine Maps
- 2.3 Constructing Affine Maps
- 2.4 Function Spaces
- 2.5 Problems
Chapter 3. Linear Interpolation
- 3.1 Linear Interpolation
- 3.2 Piecewise Linear Interpolation
- 3.3 Menelaos' Theorem
- 3.4 Blossoms
- 3.5 Barycentric Coordinates in the Plane
- 3.6 Tessellations
- 3.7 Triangulations
- 3.8 Problems
Chapter 4. The de Casteljau Algorithm
- 4.1 Parabolas
- 4.2 The de Casteljau Algorithm
- 4.3 Some Properties of Bézier Curves
- 4.4 The Blossom
- 4.5 Implementation
- 4.6 Problems
Chapter 5. The Bernstein Form of a Bézier Curve
- 5.1 Bernstein Polynomials
- 5.2 Properties of Bézier Curves
- 5.3 The Derivatives of a Bézier Curve
- 5.4 Domain Changes and Subdivision
- 5.5 Composite Bézier Curves
- 5.6 Blossom and Polar
- 5.7 The Matrix Form of a Beziér Curve
- 5.8 Implementation
- 5.9 Problems
Chapter 6. Bézier Curve Topics
- 6.1 Degree Elevation
- 6.2 Repeated Degree Elevation
- 6.3 The Variation Diminishing Property
- 6.4 Degree Reduction
- 6.5 Nonparametric Curves
- 6.6 Cross Plots
- 6.7 Integrals
- 6.8 The Bézier Form of a Bézier Curve
- 6.9 The Weierstrass Approximation Theorem
- 6.10 Formulas for Bernstein Polynomials
- 6.11 Implementation
- 6.12 Problems
Chapter 7. Polynomial Curve Constructions
- 7.1 Aitken's Algorithm
- 7.2 Lagrange Polynomials
- 7.3 The Vandermonde Approach
- 7.4 Limits of Lagrange Interpolation
- 7.5 Cubic Hermite Interpolation
- 7.6 Quintic Hermite Interpolation
- 7.7 Point-Normal Interpolation
- 7.8 Least Squares Approximation
- 7.9 Smoothing Equations
- 7.10 Designing with Bézier Curves
- 7.11 The Newton Form and Forward Differencing
- 7.12 Implementation
- 7.13 Problems
Chapter 8. B-Spline Curves
- 8.1 Motivation
- 8.2 B-Spline Segments
- 8.3 B-Spline Curves
- 8.4 Knot Insertion
- 8.5 Degree Elevation
- 8.6 Greville Abscissae
- 8.7 Smoothness
- 8.8 B-Splines
- 8.9 B-Spline Basics
- 8.10 Implementation
- 8.11 Problems
Chapter 9. Constructing Spline Curves
- 9.1 Greville Interpolation
- 9.2 Least Squares Approximation
- 9.3 Modifying B-Spline Curves
- 9.4 C2 Cubic Spline Interpolation
- 9.5 More End Conditions
- 9.6 Finding a Knot Sequence
- 9.7 The Minimum Property
- 9.8 C1 Piecewise Cubic Interpolation
- 9.9 Implementation
- 9.10 Problems
Chapter 10. W. Boehm: Differential Geometry I
- 10.1 Parametric Curves and Arc Length
- 10.2 The Frenet Frame
- 10.3 Moving the Frame
- 10.4 The Osculating Circle
- 10.5 Nonparametric Curves
- 10.6 Composite Curves
Chapter 11. Geometric Continuity
- 11.1 Motivation
- 11 2 The Direct Formulation
- 11 3 The γ, v, and β Formulations
- 11 4 C2 Cubic Splines
- 11 5 Interpolating C2 Cubic Splines
- 11.6 Higher-Order Geometric Continuity
- 11.7 Implementation
- 11.8 Problems
Chapter 12. Conic Sections
- 12.1 Projective Maps of the Real Line
- 12.2 Conies as Rational Quadratics
- 12.3 A de Casteljau Algorithm
- 12.4 Derivatives
- 12.5 The Implicit Form
- 12.6 Two Classic Problems
- 12.7 Classification
- 12.8 Control Vectors
- 12.9 Implementation
- 12.10 Problems
Chapter 13. Rational Bézier and B-Spline Curves
- 13.1 Rational Bézier Curves
- 13.2 The de Casteljau Algorithm
- 13.3 Derivatives
- 13.4 Osculatory Interpolation
- 13.5 Reparametrization and Degree Elevation
- 13.6 Control Vectors
- 13.7 Rational Cubic B-Spline Curves
- 13.8 Interpolation with Rational Cubics
- 13.9 Rational B-Splines of Arbitrary Degree
- 13.10 Implementation
- 13.11 Problems
Chapter 14. Tensor Product Patches
- 14.1 Bilinear Interpolation
- 14.2 The Direct de Casteljau Algorithm
- 14.3 The Tensor Product Approach
- 14.4 Properties
- 14.5 Degree Elevation
- 14.6 Derivatives
- 14.7 Blossoms
- 14.8 Curves on a Surface
- 14.9 Normal Vectors
- 14.10 Twists
- 14.11 The Matrix Form of a Bézier Patch
- 14.12 Nonparametric Patches
- 14.13 Problems
Chapter 15. Constructing Polynomial Patches
- 15.1 Ruled Surfaces
- 15.2 Coons Patches
- 15.3 Translational Surfaces
- 15.4 Tensor Product Interpolation
- 15.5 Bicubic Hermite Patches
- 15.6 Least Squares
- 15.7 Finding Parameter Values
- 15.8 Shape Equations
- 15.9 A Problem with Unstructured Data
- 15.10 Implementation
- 15.11 Problems
Chapter 16. Composite Surfaces
- 16.1 Smoothness and Subdivision
- 16.2 Tensor Product B-Spline Surfaces
- 16.3 Twist Estimation
- 16.4 Bicubic Spline Interpolation
- 16.5 Finding Knot Sequences
- 16.6 Rational Bézier and B-Spline Surfaces
- 16.7 Surfaces of Revolution
- 16.8 Volume Deformations
- 16.9 CONS and Trimmed Surfaces
- 16.10 Implementation
- 16.11 Problems
Chapter 17. Bézier Triangles
- 17.1 The de Casteljau Algorithm
- 17.2 Triangular Blossoms
- 17.3 Bernstein Polynomials
- 17.4 Derivatives
- 17.5 Subdivision
- 17.6 Differentiability
- 17.7 Degree Elevation
- 17.8 Nonparametric Patches
- 17.9 The Multivariate Case
- 17.10 S-Patches
- 17.11 Implementation
- 17.12 Problems
Chapter 18. Practical Aspects of Bézier Triangles
- 18.1 Rational Bézier Triangles
- 18.2 Quadrics
- 18.3 Interpolation
- 18.4 Cubic and Quintic Interpolants
- 18.5 The Clough–Tocher Interpolant
- 18.6 The Powell–Sabin Interpolant
- 18.7 Least Squares
- 18.8 Problems
Chapter 19. W. Boehm: Differential Geometry II
- 19.1 Parametric Surfaces and Arc Element
- 19.2 The Local Frame
- 19.3 The Curvature of a Surface Curve
- 19.4 Meusnier's Theorem
- 19.5 Lines of Curvature
- 19.6 Gaussian and Mean Curvature
- 19.7 Euler's Theorem
- 19.8 Dupin's Indicatrix
- 19.9 Asymptotic Lines and Conjugate Directions
- 19.10 Ruled Surfaces and Developables
- 19.11 Nonparametric Surfaces
- 19.12 Composite Surfaces
Chapter 20. Geometric Continuity for Surfaces
- 20.1 Introduction
- 20.2 Triangle-Triangle
- 20.3 Rectangle-Rectangle
- 20.4 Rectangle-Triangle
- 20.5 "Filling in" Rectangular Patches
- 20.6 "Filling in" Triangular Patches
- 20.7 Theoretical Aspects
- 20.8 Problems
Chapter 21. Surfaces with Arbitrary Topology
- 21.1 Recursive Subdivision Curves
- 21.2 Doo–Sabin Surfaces
- 21.3 Catmull–Clark Subdivision
- 21.4 Midpoint Subdivision
- 21.5 Loop Subdivision
- 21.6 √3 Subdivision
- 21.7 Interpolating Subdivision Surfaces
- 21.8 Surface Splines
- 21.9 Triangular Meshes
- 21.10 Decimation
- 21.11 Problems
Chapter 22. Coons Patches
- 22.1 Coons Patches: Bilinearly Blended
- 22.2 Coons Patches: Partially Bicubically Blended
- 22.3 Coons Patches: Bicubically Blended
- 22.4 Piecewise Coons Surfaces
- 22.5 Two Properties
- 22.6 Compatibility
- 22.7 Gordon Surfaces
- 22.8 Boolean Sums
- 22.9 Triangular Coons Patches
- 22.10 Problems
Chapter 23. Shape
- 23.1 Use of Curvature Plots
- 23.2 Curve and Surface Smoothing
- 23.3 Surface Interrogation
- 23.4 Implementation
- 23.5 Problems
Chapter 24. Evaluation of Some Methods
- 24.1 Bézier Curves or B-Spline Curves?
- 24.2 Spline Curves or B-Spline Curves?
- 24.3 The Monomial or the Bézier Form?
- 24.4 The B-Spline or the Hermite Form?
- 24.5 Triangular or Rectangular Patches?
Appendix A. Quick Reference of Curve and Surface Terms
Appendix B. List of Programs
Appendix C. Notation
References
Index
Color Plate Section

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