Curves and Surfaces for Computer-Aided Geometric Design

Transcription

1 Curves and Surfaces for Computer-Aided Geometric Design A Practical Guide Fourth Edition Gerald Farin Department of Computer Science Arizona State University Tempe, Arizona /ACADEMIC PRESS I San Diego London Boston.NewYork?Sydney Tokyo Toronto

2 Preface xv 1 P. Bezier: How a Simple System Was Born 1 2 Introductory Material Points and Vectors Affine Maps Linear Interpolation Piecewise Linear Interpolation Menelaos' Theorem Barycentric Coordinates in the Plane Tessellations and Triangulations Function Spaces Exercises 31 The de Casteljau Algorithm 3.1 Parabolas 3.2 The de Casteljau Algorithm 3.3 Some Properties of Bezier Curves 3.4 The Blossom 3.5 Implementation 3.6 Exercises The Bernstein Form of a Bezier Curve 4.1 Bernstein Polynomials 4.2' Properties of Bezier Curves 4.3 The Derivative of a Bezier Curve 4.4 Higher Order Derivatives 4.5 Derivatives and the de Casteljau Algorithm 4.6 Subdivision 4.7 Blossom and Polar 4.8 The Matrix Form of a Bezier Curve 4.9 Implementation 4.10 Exercises

3 Bezier Curve Topics Degree Elevation Repeated Degree Elevation The Variation Diminishing Property Degree Reduction Nonparametric Curves Cross Plots Integrals The Bezier Form of a Bezier Curve The Barycentric Form of a Bezier Curve The Weierstrass Approximation Theorem Formulas for Bernstein Polynomials Implementation Exercises 79 Polynomial Interpolation 6.1 Aitken's Algorithm 6.2 Lagrange Polynomials 6.3 The Vandermonde Approach 6.4 Limits of Lagrange Interpolation 6.5 Cubic Hermite Interpolation 6.6 Quintic Hermite Interpolation 6.7 The Newton Form and Forward Differencing 6.8 Implementation 6.9 Exercises Spline Curves in Bezier Form Global and Local Parameters Smoothness Conditions C 1 and C 2 Continuity Finding a C 1 Parametrization C 1 Quadratic B-spline Curves C 2 Cubic B-spline Curves Finding a Knot Sequence Design and Inverse Design Implementation Exercises 112 Piecewise Cubic Interpolation C 1 Piecewise Cubic Hermite Interpolation C 1 Piecewise Cubic Interpolation I C 1 Piecewise Cubic Interpolation II Point-Normal Interpolation Font Design Exercises

4 Cubic Spline Interpolation The B-spline Form The Hermite Form End Conditions Finding a Knot Sequence The Minimum Property Implementation Exercises B-splines 10.1 Motivation 10.2 Knot Insertion 10.3 The de Boor Algorithm 10.4 Smoothness of B-spline Curves 10.5 The B-spline Basis 10.6 Two Recursion Formulas 10.7 Repeated Knot Insertion 10.8 B-spline Properties 10.9 B-spline Blossoms Approximation B-spline Basics Implementation Exercises 11 W. Boehm: Differential Geometry I 11.1 Parametric Curves and Arc Length 11.2 The Frenet Frame Moving the Frame 11.4 The Osculating Circle 11.5 Nonparametric Curves 11.6 Composite Curves Geometric Continuity Motivation The Direct Formulation The 7 Formulation The v and j3 Formulation Comparison G 2 Cubic Splines Interpolating G 2 Cubic Splines Local Basis Functions for G 2 Splines Higher Order Geometric Continuity Implementation Exercises 195

5 x Contents 13 Conic Sections Projective Maps of the Real Line Conies as Rational Quadratics A de Casteljau Algorithm Derivatives The Implicit Form Two Classic Problems Classification Control Vectors Implementation Exercises Rational Bezier and B-spline Curves Rational Bezier Curves The de Casteljau Algorithm Derivatives Osculatory Interpolation Reparametrization and Degree Elevation Control Vectors Rational Cubic B-spline Curves Interpolation with Rational Cubics Rational B-splines of Arbitrary Degree Implementation Exercises Tensor Product Patches Bilinear Interpolation The Direct de Casteljau Algorithm The Tensor Product Approach Properties Degree Elevation Derivatives Blossoms Normal Vectors Twists The Matrix Form of a Bezier Patch Nonparametric Patches Tensor Product Interpolation Bicubic Hermite Patches Implementation Exercises Composite Surfaces and Spline Interpolation Smoothness and Subdivision Tensor Product B-spline Surfaces 258

6 xi 16.3 Twist Estimation Bicubic Spline Interpolation Finding Knot Sequences Rational Bezier and B-spline Surfaces Surfaces of Revolution Volume Deformations CONS and Trimmed Surfaces Implementation ' Exercises Bezier Triangles The de Casteljau Algorithm Triangular Blossoms Bernstein Polynomials Derivatives Subdivision Differentiability Degree Elevation Nonparametric Patches Rational Bezier Triangles Quadrics Interpolation Cubic and Quintic Interpolants The Clough-Tocher Interpolant The Powell-Sabin Interpolant Implementation Exercises Geometric Continuity for Surfaces Introduction Triangle-Triangle Rectangle-Rectangle Rectangle-Triangle "Filling In" Rectangular Patches "Filling In" Triangular Patches Theoretical Aspects Exercises Surfaces with Arbitrary Topology Doo-Sabin Surfaces 317 ' 19.2 Interpolation S-Patches Surface Splines Exercises 324

7 xii Contents 20 Coons Patches Ruled Surfaces Coons Patches: Bilinearly Blended " Coons Patches: Partially Bicubically Blended Coons Patches: Bicubically Blended Piecewise Coons Surfaces Exercises Coons Patches: Additional Material Compatibility Control Nets from Coons Patches Translational Surfaces Gordon Surfaces Boolean Sums Triangular Coons Patches Implementation Exercises W. Boehm: Differential Geometry II Parametric Surfaces and Arc Element The Local Frame The Curvature of a Surface Curve Meusnier's Theorem Lines of Curvature Gaussian and Mean Curvature Euler's Theorem Dupin's Indicatrix Asymptotic Lines and ConjugateDirections Ruled Surfaces and Developables Nonparametric Surfaces Composite Surfaces Interrogation and Smoothing Use of Curvature Plots Curve and Surface Smoothing Surface Interrogation Implementation Exercises Evaluation of Some Methods Bezier Curves or B-spline Curves? Spline Curves or B-spline Curves? The Monomial or the Bezier Form? The B-spline or the Hermite Form? Triangular or Rectangular Patches? 376

8 xiii 25 Quick Reference of Curve and Surface Terms 378 Appendix 1: List of Programs 385 Appendix 2: Notation 386 Bibliography 387 Index 421