Farin Curves and Surfaces for Computer-Aided Geometric Design

A Practical Guide
3. Auflage 2014
ISBN: 978-1-4832-9699-9
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 473 Seiten, Web PDF

A leading expert in CAGD, Gerald Farin covers the representation, manipulation, and evaluation of geometric shapes in this the Third Edition of Curves and Surfaces for Computer Aided Geometric Design. The book offers an introduction to the field that emphasizes Bernstein-Bezier methods and presents subjects in an informal, readable style, making this an ideal text for an introductory course at the advanced undergraduate or graduate level. The Third Edition includes a new chapter on Topology, offers new exercises and sections within most chapters, combines the material on Geometric Continuity into one chapter, and updates existing materials and references. Implementation techniques are addressed for practitioners by the inclusion of new C programs for many of the fundamental algorithms. The C programs are available on a disk included with the text. System Requirements: IBM PC or compatibles, DOS version 2.0 or higher. - Covers representation, manipulation, and evaluation of geometric shapes - Emphasizes Bernstein-Bezier methods - Written in an informal, easy-to-read style

Professor Gerald Farin currently teaches in the computer science and engineering department at Arizona State University. He received his doctoral degree in mathematics from the University of Braunschweig, Germany, in 1979. His extensive CAGD experience includes working as a research mathematician in a computer-aided development for Daimler-Benz, serving on the executive committee of the ASU PRISM project, and speaking at a multitude of symposia and conferences. Farin has authored and edited several books and papers, and he is editor-in-chief of Computer Aided Geometric Design.

Autoren/Hrsg.

Farin, Gerald

Inhaltsverzeichnis

1;Front Cover;1
2;Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide;4
3;Copyright Page;5
4;Table of Contents;8
5;Preface;16
6;Chapter 1. P. Bézier: How a Simple System Was Born;20
7;Chapter 2. Introductory Material;32
7.1;2.1 Points and Vectors;32
7.2;2.2 Affine Maps;36
7.3;2.3 Linear Interpolation;39
7.4;2.4 Piecewise Linear Interpolation;42
7.5;2.5 Menelaos' Theorem;43
7.6;2.6 Function Spaces;44
7.7;2.7 Problems;46
8;Chapter 3. The de Casteljau Algorithm;48
8.1;3.1 Parabolas;48
8.2;3.2 The de Casteljau Algorithm;50
8.3;3.3 Some Properties of Bézier Curves;51
8.4;3.4 The Blossom;55
8.5;3.5 Implementation;58
8.6;3.6 Problems;59
9;Chapter 4. The Bernstein Form of a Bézier Curve;60
9.1;4.1 Bernstein Polynomials;60
9.2;4.2 Properties of Bézier Curves;63
9.3;4.3 The Derivative of a Bézier Curve;65
9.4;4.4 Higher Order Derivatives;66
9.5;4.5 Derivatives and the de Casteljau Algorithm;69
9.6;4.6 Subdivision;70
9.7;4.7 Blossom and Polar;75
9.8;4.8 The Matrix Form of a Bézier Curve;77
9.9;4.9 Implementation;78
9.10;4.10 Problems;81
10;Chapter 5. Bézier Curve Topics;84
10.1;5.1 Degree Elevation;84
10.2;5.2 Repeated Degree Elevation;85
10.3;5.3 The Variation Diminishing Property;86
10.4;5.4 Degree Reduction;87
10.5;5.5 Nonparametric Curves;89
10.6;5.6 Cross Plots;91
10.7;5.7 Integrals;91
10.8;5.8 The Bézier Form of a Bézier Curve;93
10.9;5.9 The Barycentric Form of a Bézier Curve;94
10.10;5.10 The Weierstrass Approximation Theorem;97
10.11;5.11 Formulas for Bernstein Polynomials;98
10.12;5.12 Implementation;99
10.13;5.13 Problems;100
11;Chapter 6. Polynomial Interpolation;102
11.1;6.1 Aitken's Algorithm;102
11.2;6.2 Lagrange Polynomials;105
11.3;6.3 The Vandermonde Approach;107
11.4;6.4 Limits of Lagrange Interpolation;108
11.5;6.5 Cubic Hermite Interpolation;110
11.6;6.6 Quintic Hermite Interpolation;115
11.7;6.7 The Newton Form and Forward Differencing;115
11.8;6.8 Implementation;118
11.9;6.9 Problems;118
12;Chapter 7. Spline Curves in Bézier Form;120
12.1;7.1 Global and Local Parameters;120
12.2;7.2 Smoothness Conditions;122
12.3;7.3 C1 Continuity;124
12.4;7.4 C2 Continuity;125
12.5;7.5 Finding a C1 Parametrization;126
12.6;7.6 C1 Quadratic B-spline Curves;128
12.7;7.7 C2 Cubic B-spline Curves;134
12.8;7.8 Parametrizations;136
12.9;7.9 Design and Inverse Design;137
12.10;7.10 Implementation;138
12.11;7.11 Problems;139
13;Chapter 8. Piecewise Cubic Interpolation;140
13.1;8.1 C1 Piecewise Cubic Hermite Interpolation;140
13.2;8.2 C1 Piecewise Cubic Interpolation I;142
13.3;8.3 C1 Piecewise Cubic Interpolation II;145
13.4;8.4 Point-Normal Interpolation;148
13.5;8.5 Font Generation;149
13.6;8.6 Problems;149
14;Chapter 9. Cubic Spline Interpolation;152
14.1;9.1 The B-spline Form;152
14.2;9.2 The Hermite Form;156
14.3;9.3 End Conditions;158
14.4;9.4 Parametrization;163
14.5;9.5 The Minimum Property;167
14.6;9.6 Implementation;171
14.7;9.7 Problems;173
15;Chapter 10. B-splines;176
15.1;10.1 Motivation;177
15.2;10.2 Knot Insertion;178
15.3;10.3 The de Boor Algorithm;184
15.4;10.4 Smoothness of B-spline Curves;187
15.5;10.5 The B-spline Basis;188
15.6;10.6 Two Recursion Formulas;191
15.7;10.7 Repeated Knot Insertion;194
15.8;10.8 Additional Material;197
15.9;10.9 B-spline Blossoms;200
15.10;10.10 B-spline Basics;203
15.11;10.11 Implementation;204
15.12;10.12 Problems;205
16;Chapter 11. W. Boehm: Differential Geometry I;208
16.1;11.1 Parametric Curves and Arc Length;208
16.2;11.2 The Frenet Frame;210
16.3;11.3 Moving the Frame;211
16.4;11.4 The Osculating Circle;213
16.5;11.5 Nonparametric Curves;216
16.6;11.6 Composite Curves;216
17;Chapter 12. Geometric Continuity I;220
17.1;12.1 Motivation;220
17.2;12.2 A Characterization of G2 Curves;221
17.3;12.3 Nu-splines;223
17.4;12.4 G2 Piecewise Bézier Curves;226
17.5;12.5 Direct G2 Cubic Splines;229
17.6;12.6 Implementation;231
17.7;12.7 Problems;232
18;Chapter 13. Geometric Continuity II;234
18.1;13.1 Gamma-splines;234
18.2;13.2 Local Basis Functions for G2 Splines;237
18.3;13.3 Beta-splines;241
18.4;13.4 Higher Order Geometric Continuity;246
18.5;13.5 Implementation;249
18.6;13.6 Problems;250
19;Chapter 14. Conic Sections;252
19.1;14.1 Projective Maps of the Real Line;252
19.2;14.2 Conies as Rational Quadratics;256
19.3;14.3 A de Casteljau Algorithm;261
19.4;14.4 Derivatives;262
19.5;14.5 The Implicit Form;263
19.6;14.6 Two Classic Problems;265
19.7;14.7 Classification;267
19.8;14.8 Control Vectors;269
19.9;14.9 Implementation;271
19.10;14.10 Problems;272
20;Chapter 15. Rational Bézier and B-spline Curves;274
20.1;15.1 Rational Bézier Curves;274
20.2;15.2 The de Casteljau Algorithm;275
20.3;15.3 Derivatives;278
20.4;15.4 Osculatory Interpolation;279
20.5;15.5 Reparametrization and Degree Elevation;280
20.6;15.6 Control Vectors;283
20.7;15.7 Rational Cubic B-spline Curves;284
20.8;15.8 Interpolation with Rational Cubics;286
20.9;15.9 Rational B-splines of Arbitrary Degree;287
20.10;15.10 Implementation;288
20.11;15.11 Problems;289
21;Chapter 16. Tensor Product Bézier Surfaces;290
21.1;16.1 Bilinear Interpolation;290
21.2;16.2 The Direct de Casteljau Algorithm;292
21.3;16.3 The Tensor Product Approach;294
21.4;16.4 Properties;300
21.5;16.5 Degree Elevation;301
21.6;16.6 Derivatives;302
21.7;16.7 Normal Vectors;304
21.8;16.8 Twists;307
21.9;16.9 The Matrix Form of a Bézier Patch;308
21.10;16.10 Nonparametric Patches;309
21.11;16.11 Implementation;311
21.12;16.12 Problems;311
22;Chapter 17. Composite Surfaces and Spline Interpolation;314
22.1;17.1 Smoothness and Subdivision;314
22.2;17.2 Bicubic B-spline Surfaces;317
22.3;17.3 Twist Estimation;319
22.4;17.4 Tensor Product Interpolants;324
22.5;17.5 The Parametrization;328
22.6;17.6 Bicubic Hermite Patches;330
22.7;17.7 Rational Bézier and B-spline Surfaces;332
22.8;17.8 Surfaces of Revolution;334
22.9;17.9 Volume Deformations;336
22.10;17.10 Trimmed Surfaces;339
22.11;17.11 Implementation;340
22.12;17.12 Problems;342
23;Chapter 18. Bézier Triangles;344
23.1;18.1 Barycentric Coordinates and Linear Interpolation;344
23.2;18.2 The de Casteljau Algorithm;347
23.3;18.3 Triangular Blossoms;351
23.4;18.4 Bernstein Polynomials;352
23.5;18.5 Derivatives;354
23.6;18.6 Subdivision;358
23.7;18.7 Differentiability;361
23.8;18.8 Degree Elevation;362
23.9;18.9 Nonparametric Patches;363
23.10;18.10 Rational Bézier Triangles;366
23.11;18.11 Quadrics;369
23.12;18.12 Implementation;373
23.13;18.13 Problems;373
24;Chapter 19. Geometric Continuity for Surfaces;376
24.1;19.1 Introduction;376
24.2;19.2 Triangle-Triangle;377
24.3;19.3 Rectangle-Rectangle;381
24.4;19.4 Rectangle-Triangle;382
24.5;19.5 "Filling in" Rectangular Patches;383
24.6;19.6 "Filling in" Triangular Patches;384
24.7;19.7 Theoretical Aspects;385
24.8;19.8 Problems;385
25;Chapter 20. Coons Patches;386
25.1;20.1 Ruled Surfaces;387
25.2;20.2 Coons Patches: Bilinearly Blended;388
25.3;20.3 Coons Patches: Partially Bicubically Blended;391
25.4;20.4 Coons Patches: Bicubically Blended;393
25.5;20.5 Piecewise Coons Surfaces;395
25.6;20.6 Problems;396
26;Chapter 21. Coons Patches: Additional Material;398
26.1;21.1 Compatibility;398
26.2;21.2 Control Nets from Coons Patches;401
26.3;21.3 Translational Surfaces;403
26.4;21.4 Gordon Surfaces;404
26.5;21.5 Boolean Sums;406
26.6;21.6 Triangular Coons Patches;408
26.7;21.7 Implementation;411
26.8;21.8 Problems;411
27;Chapter 22. W. Boehm: Differential Geometry II;412
27.1;22.1 Parametric Surfaces and Arc Element;412
27.2;22.2 The Local Frame;414
27.3;22.3 The Curvature of a Surface Curve;415
27.4;22.4 Meusnier's Theorem;417
27.5;22.5 Lines of Curvature;418
27.6;22.6 Gaussian and Mean Curvature;420
27.7;22.7 Euler's Theorem;421
27.8;22.8 Dupin's Indicatrix;422
27.9;22.9 Asymptotic Lines and Conjugate Directions;423
27.10;22.10 Ruled Surfaces and Developables;424
27.11;22.11 Nonparametric Surfaces;426
27.12;22.12 Composite Surfaces;427
28;Chapter 23. Interrogation and Smoothing;430
28.1;23.1 Use of Curvature Plots;430
28.2;23.2 Curve and Surface Smoothing;431
28.3;23.3 Surface Interrogation;434
28.4;23.4 Implementation;437
28.5;23.5 Problems;439
29;Chapter 24. Evaluation of Some Methods;440
29.1;24.1 Bézier Curves or B-spline Curves?;440
29.2;24.2 Spline Curves or B-spline Curves?;440
29.3;24.3 The Monomial or the Bézier Form?;441
29.4;24.4 The B-spline or the Hermite Form?;444
29.5;24.5 Triangular or Rectangular Patches?;445
30;Chapter 25. Quick Reference of Curve and Surface Terms;448
31;Appendix 1: List of Programs;454
32;Appendix 2: Notation;456
33;Bibliography;458
34;Index;488