Front Cover -- Curves and Surfaces for CAGD: A Practical Guide -- 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, ν, 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.