Curves and Surfaces for Computer-Aided Geometric Design (eBook)

Front Cover 1
Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide 4
Copyright Page 5
Table of Contents 8
Preface 16
Chapter 1. P. Bézier: How a Simple System Was Born 20
Chapter 2. Introductory Material 32
2.1 Points and Vectors 32
2.2 Affine Maps 36
2.3 Linear Interpolation 39
2.4 Piecewise Linear Interpolation 42
2.5 Menelaos' Theorem 43
2.6 Function Spaces 44
2.7 Problems 46
Chapter 3. The de Casteljau Algorithm 48
3.1 Parabolas 48
3.2 The de Casteljau Algorithm 50
3.3 Some Properties of Bézier Curves 51
3.4 The Blossom 55
3.5 Implementation 58
3.6 Problems 59
Chapter 4. The Bernstein Form of a Bézier Curve 60
4.1 Bernstein Polynomials 60
4.2 Properties of Bézier Curves 63
4.3 The Derivative of a Bézier Curve 65
4.4 Higher Order Derivatives 66
4.5 Derivatives and the de Casteljau Algorithm 69
4.6 Subdivision 70
4.7 Blossom and Polar 75
4.8 The Matrix Form of a Bézier Curve 77
4.9 Implementation 78
4.10 Problems 81
Chapter 5. Bézier Curve Topics 84
5.1 Degree Elevation 84
5.2 Repeated Degree Elevation 85
5.3 The Variation Diminishing Property 86
5.4 Degree Reduction 87
5.5 Nonparametric Curves 89
5.6 Cross Plots 91
5.7 Integrals 91
5.8 The Bézier Form of a Bézier Curve 93
5.9 The Barycentric Form of a Bézier Curve 94
5.10 The Weierstrass Approximation Theorem 97
5.11 Formulas for Bernstein Polynomials 98
5.12 Implementation 99
5.13 Problems 100
Chapter 6. Polynomial Interpolation 102
6.1 Aitken's Algorithm 102
6.2 Lagrange Polynomials 105
6.3 The Vandermonde Approach 107
6.4 Limits of Lagrange Interpolation 108
6.5 Cubic Hermite Interpolation 110
6.6 Quintic Hermite Interpolation 115
6.7 The Newton Form and Forward Differencing 115
6.8 Implementation 118
6.9 Problems 118
Chapter 7. Spline Curves in Bézier Form 120
7.1 Global and Local Parameters 120
7.2 Smoothness Conditions 122
7.3 C1 Continuity 124
7.4 C2 Continuity 125
7.5 Finding a C1 Parametrization 126
7.6 C1 Quadratic B-spline Curves 128
7.7 C2 Cubic B-spline Curves 134
7.8 Parametrizations 136
7.9 Design and Inverse Design 137
7.10 Implementation 138
7.11 Problems 139
Chapter 8. Piecewise Cubic Interpolation 140
8.1 C1 Piecewise Cubic Hermite Interpolation 140
8.2 C1 Piecewise Cubic Interpolation I 142
8.3 C1 Piecewise Cubic Interpolation II 145
8.4 Point-Normal Interpolation 148
8.5 Font Generation 149
8.6 Problems 149
Chapter 9. Cubic Spline Interpolation 152
9.1 The B-spline Form 152
9.2 The Hermite Form 156
9.3 End Conditions 158
9.4 Parametrization 163
9.5 The Minimum Property 167
9.6 Implementation 171
9.7 Problems 173
Chapter 10. B-splines 176
10.1 Motivation 177
10.2 Knot Insertion 178
10.3 The de Boor Algorithm 184
10.4 Smoothness of B-spline Curves 187
10.5 The B-spline Basis 188
10.6 Two Recursion Formulas 191
10.7 Repeated Knot Insertion 194
10.8 Additional Material 197
10.9 B-spline Blossoms 200
10.10 B-spline Basics 203
10.11 Implementation 204
10.12 Problems 205
Chapter 11. W. Boehm: Differential Geometry I 208
11.1 Parametric Curves and Arc Length 208
11.2 The Frenet Frame 210
11.3 Moving the Frame 211
11.4 The Osculating Circle 213
11.5 Nonparametric Curves 216
11.6 Composite Curves 216
Chapter 12. Geometric Continuity I 220
12.1 Motivation 220
12.2 A Characterization of G2 Curves 221
12.3 Nu-splines 223
12.4 G2 Piecewise Bézier Curves 226
12.5 Direct G2 Cubic Splines 229
12.6 Implementation 231
12.7 Problems 232
Chapter 13. Geometric Continuity II 234
13.1 Gamma-splines 234
13.2 Local Basis Functions for G2 Splines 237
13.3 Beta-splines 241
13.4 Higher Order Geometric Continuity 246
13.5 Implementation 249
13.6 Problems 250
Chapter 14. Conic Sections 252
14.1 Projective Maps of the Real Line 252
14.2 Conies as Rational Quadratics 256
14.3 A de Casteljau Algorithm 261
14.4 Derivatives 262
14.5 The Implicit Form 263
14.6 Two Classic Problems 265
14.7 Classification 267
14.8 Control Vectors 269
14.9 Implementation 271
14.10 Problems 272
Chapter 15. Rational Bézier and B-spline Curves 274
15.1 Rational Bézier Curves 274
15.2 The de Casteljau Algorithm 275
15.3 Derivatives 278
15.4 Osculatory Interpolation 279
15.5 Reparametrization and Degree Elevation 280
15.6 Control Vectors 283
15.7 Rational Cubic B-spline Curves 284
15.8 Interpolation with Rational Cubics 286
15.9 Rational B-splines of Arbitrary Degree 287
15.10 Implementation 288
15.11 Problems 289
Chapter 16. Tensor Product Bézier Surfaces 290
16.1 Bilinear Interpolation 290
16.2 The Direct de Casteljau Algorithm 292
16.3 The Tensor Product Approach 294
16.4 Properties 300
16.5 Degree Elevation 301
16.6 Derivatives 302
16.7 Normal Vectors 304
16.8 Twists 307
16.9 The Matrix Form of a Bézier Patch 308
16.10 Nonparametric Patches 309
16.11 Implementation 311
16.12 Problems 311
Chapter 17. Composite Surfaces and Spline Interpolation 314
17.1 Smoothness and Subdivision 314
17.2 Bicubic B-spline Surfaces 317
17.3 Twist Estimation 319
17.4 Tensor Product Interpolants 324
17.5 The Parametrization 328
17.6 Bicubic Hermite Patches 330
17.7 Rational Bézier and B-spline Surfaces 332
17.8 Surfaces of Revolution 334
17.9 Volume Deformations 336
17.10 Trimmed Surfaces 339
17.11 Implementation 340
17.12 Problems 342
Chapter 18. Bézier Triangles 344
18.1 Barycentric Coordinates and Linear Interpolation 344
18.2 The de Casteljau Algorithm 347
18.3 Triangular Blossoms 351
18.4 Bernstein Polynomials 352
18.5 Derivatives 354
18.6 Subdivision 358
18.7 Differentiability 361
18.8 Degree Elevation 362
18.9 Nonparametric Patches 363
18.10 Rational Bézier Triangles 366
18.11 Quadrics 369
18.12 Implementation 373
18.13 Problems 373
Chapter 19. Geometric Continuity for Surfaces 376
19.1 Introduction 376
19.2 Triangle-Triangle 377
19.3 Rectangle-Rectangle 381
19.4 Rectangle-Triangle 382
19.5 "Filling in" Rectangular Patches 383
19.6 "Filling in" Triangular Patches 384
19.7 Theoretical Aspects 385
19.8 Problems 385
Chapter 20. Coons Patches 386
20.1 Ruled Surfaces 387
20.2 Coons Patches: Bilinearly Blended 388
20.3 Coons Patches: Partially Bicubically Blended 391
20.4 Coons Patches: Bicubically Blended 393
20.5 Piecewise Coons Surfaces 395
20.6 Problems 396
Chapter 21. Coons Patches: Additional Material 398
21.1 Compatibility 398
21.2 Control Nets from Coons Patches 401
21.3 Translational Surfaces 403
21.4 Gordon Surfaces 404
21.5 Boolean Sums 406
21.6 Triangular Coons Patches 408
21.7 Implementation 411
21.8 Problems 411
Chapter 22. W. Boehm: Differential Geometry II 412
22.1 Parametric Surfaces and Arc Element 412
22.2 The Local Frame 414
22.3 The Curvature of a Surface Curve 415
22.4 Meusnier's Theorem 417
22.5 Lines of Curvature 418
22.6 Gaussian and Mean Curvature 420
22.7 Euler's Theorem 421
22.8 Dupin's Indicatrix 422
22.9 Asymptotic Lines and Conjugate Directions 423
22.10 Ruled Surfaces and Developables 424
22.11 Nonparametric Surfaces 426
22.12 Composite Surfaces 427
Chapter 23. Interrogation and Smoothing 430
23.1 Use of Curvature Plots 430
23.2 Curve and Surface Smoothing 431
23.3 Surface Interrogation 434
23.4 Implementation 437
23.5 Problems 439
Chapter 24. Evaluation of Some Methods 440
24.1 Bézier Curves or B-spline Curves? 440
24.2 Spline Curves or B-spline Curves? 440
24.3 The Monomial or the Bézier Form? 441
24.4 The B-spline or the Hermite Form? 444
24.5 Triangular or Rectangular Patches? 445
Chapter 25. Quick Reference of Curve and Surface Terms 448
Appendix 1: List of Programs 454
Appendix 2: Notation 456
Bibliography 458
Index 488