Lyche / Schumaker Mathematical Methods in Computer Aided Geometric Design

1. Auflage 2014
ISBN: 978-1-4832-5780-8
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 628 Seiten, Web PDF

Mathematical Methods in Computer Aided Geometric Design covers the proceedings of the 1988 International Conference by the same title, held at the University of Oslo, Norway. This text contains papers based on the survey lectures, along with 33 full-length research papers. This book is composed of 39 chapters and begins with surveys of scattered data interpolation, spline elastic manifolds, geometry processing, the properties of B‚zier curves, and Gr”bner basis methods for multivariate splines. The next chapters deal with the principles of box splines, smooth piecewise quadric surfaces, some applications of hierarchical segmentations of algebraic curves, nonlinear parameters of splines, and algebraic aspects of geometric continuity. These topics are followed by discussions of shape preserving representations, box-spline surfaces, subdivision algorithm parallelization, interpolation systems, and the finite element method. Other chapters explore the concept and applications of uniform bivariate hermite interpolation, an algorithm for smooth interpolation, and the three B-spline constructions. The concluding chapters consider the three B-spline constructions, design tools for shaping spline models, approximation of surfaces constrained by a differential equation, and a general subdivision theorem for B‚zier triangles. This book will prove useful to mathematicians and advance mathematics students.

Autoren/Hrsg.

Lyche, Tom

Schumaker, Larry L.

Inhaltsverzeichnis

1;Front Cover;1
2;Mathematical Methods in Computer Aided Geometric Design;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE;9
6;PARTICIPANTS;10
7;Chapter 1. Scattered Data Interpolation in Three or More Variables;18
7.1;§1. Introduction;18
7.2;§2. Rendering of Trivariate Functions;24
7.3;§3. Tensor Product Schemes;25
7.4;§4. Point Schemes;26
7.5;§5. Natural Neighbor Interpolation;29
7.6;§6. k-dimensional Triangulations;30
7.7;§7. Tetrahedral Schemes;32
7.8;§8. Simplicial Schemes;36
7.9;§9. Multivariate Splines;39
7.10;§10. Transfinite Hypercubal Methods;42
7.11;§11. Derivative Generation;42
7.12;§12. Interpolation on the sphere and other surfaces;44
7.13;§13. Conclusions;44
7.14;Acknowledgments;46
7.15;References;46
8;Chapter 2. Some Applications of Discrete Dm Splines;52
8.1;§1. Vh-Discrete Smoothing Dm-Splines;52
8.2;§3. Approximation of non-Regular Functions;55
8.3;References;60
9;Chapter 3. Spline Elastic Manifolds;62
9.1;§1. Introduction;62
9.2;§2. The Space k;63
9.3;§3. The Hilbert Kernel of k;65
9.4;§4. Characterization of the elastic spline manifold;67
9.5;References;67
10;Chapter 4. Geometry Processing: Curvature Analysis and Surface-Surface Intersection;68
10.1;§1. Introduction to Geometry Processing;68
10.2;§2. Curve Fairing;69
10.3;§3. Surface Curvature Analysis;69
10.4;§4. Surface-Surface Intersection;72
10.5;§5. Offset Surfaces;74
10.6;Acknowledgments;76
10.7;References;76
11;Chapter 5. Three Examples of Dual Properties of Bézier Curves;78
11.1;§1. Introduction;78
11.2;§2. Example 1. Degree Elevation and Differentiation;79
11.3;§3. Example 2. Transformations to and from Monomial Form;81
11.4;§4. Example 3. de Casteljau and Horner Evaluation;82
11.5;§5. Concluding Remarks;86
11.6;References;86
12;Chapter 6. What is the Natural Generalization of a Bézier Curve?;88
12.1;§1. Introduction;88
12.2;§2. The Canonical Split;89
12.3;§3. B-Spline Properties;92
12.4;§4. Pólya Properties;94
12.5;§5. Shared Properties and Dual Properties;98
12.6;§6. Conclusions;100
12.7;References;101
13;Chapter 7. Convexity and a Multidimensional Version of the Variation Diminishing Property of Bernstein Polynomials;104
13.1;§1. Notation and Definitions;104
13.2;§2. Piecewise Linear Surface Over a Convex Polyhedron;105
13.3;§3. Variation Diminishing Property of Bernstein Polynomials;107
13.4;Acknowledgment;109
13.5;References;109
14;Chapter 8. Gröbner Basis Methods for Multivariate Splines;110
14.1;§1. Dimensions of Spline Spaces;110
14.2;§2. Gröbner Bases;113
14.3;§3. Computing Dimension Series and Bases of Splines;116
14.4;§4. Example and Discussion;117
14.5;References;119
15;Chapter 9. On Finite Element Interpolation Problems;122
15.1;§1. Introduction;122
15.2;§2. Interpolation Systems in IR;122
15.3;§3. Interpolation Problem Associated to an Interpolation System;123
15.4;§4. Argyris Triangle;124
15.5;§5. Construction of the Solution of the Interpolation Problem;126
15.6;§6. Basic Functions for the Argyris Triangle;128
15.7;References;129
16;Chapter 10. The Design of Curves and Surfaces by Subdivision Algorithms;132
16.1;§1. Introduction;132
16.2;§2. The Algorithms of de Casteljau and Chaikin;133
16.3;§3. De Rham's Construction of Certain Planar Curves;138
16.4;§4. Algorithms for Surfaces;140
16.5;§5. The Subdivision Algorithm for Bernstein-Bézier Curves;141
16.6;§6. Subdivision Algorithms for Univariate Spline Functions;144
16.7;§7. Cube Splines and the Line Average Algorithm;148
16.8;§8. Rates of Convergence;154
16.9;§9. Subdivision as Corner Cutting;156
16.10;§10. A Matrix Approach to Subdivision for the Univariate Case;158
16.11;§11. Regular Subdivision;162
16.12;References;167
17;Chapter 11. A Data Dependent Parametrization for Spline Approximation;172
17.1;§1. Introduction;172
17.2;§2. Background;174
17.3;§3. Finding an Initial Parametrization;176
17.4;§4. Experimental Results;180
17.5;§5. Remarks;182
17.6;References;182
18;Chapter 12. On the Evaluation of Box Splines;184
18.1;§1. Introduction;184
18.2;§2. Boxes, Cones and Simplex Splines;185
18.3;§3. Regular Meshes;186
18.4;§4. The Bivariate Case;187
18.5;§5. The Trivariate Case;191
18.6;Acknowledgment;195
18.7;References;195
19;Chapter 13. Smooth Piecewise Quadric Surfaces;198
19.1;§1. Introduction;198
19.2;§2. Topology of Interpolating Surfaces and Transversal Systems;199
19.3;§3. Implicit Bézier Patches;201
19.4;§4. Macro Patches for Piecewise Quadric Surfaces;202
19.5;References;210
20;CHapter 14. Inserting New Knots Into Beta-Spline Curves;212
20.1;§1. Introduction;212
20.2;§2. Generalized Cubic Splines and ß-splines;214
20.3;§3. Knot Insertion;216
20.4;§4. Applications in CAGD;218
20.5;References;221
21;Chapter 15. Recursive Subdivision and Iteration in Intersections and Related Problems;224
21.1;§1. Introduction;224
21.2;§2. Comparison of Iteration and Recursive Subdivision;225
21.3;§3. General Layout of the Intersection Strategy;227
21.4;§4. Critical Subalgorithms;229
21.5;§5. Conclusion;231
21.6;References;231
22;Chapter 16. Rational Curves and Surfaces;232
22.1;§1. Introduction;232
22.2;§2. Conies as Rational Quadratics;232
22.3;§3. Derivatives;235
22.4;§4. Classification;236
22.5;§6. Rational Bezier Curves;238
22.6;§7. The de Casteljau Algorithm;239
22.7;§8. Derivatives;241
22.8;§9. Reparametrization and Degree Elevation;242
22.9;§10. Functional Rational Bézier Curves;244
22.10;§11. Rational Cubic B-spline Curves;245
22.11;§12. Rational B-splines of Arbitrary Degree;247
22.12;§13. Reparametrizations of Rational B-spline Curves;248
22.13;§14. Interpolation with Rational Cubics;248
22.14;§15. Rational B-spline Surfaces;250
23;Chapter 17. Algebraic Aspects of Geometric Continuity;334
23.1;§1. Introduction;334
23.2;§2. Some Algebraic Aspects of Geometric Continuity;335
23.3;§4. The Invariance of Frenet Frame Continuity Under Projection;342
23.4;§5. Some Further Algebraic Aspects of Geometric Continuity;347
23.5;§6. Conclusion;352
23.6;References;352
24;Chapter 18. Shape Preserving Representations;354
24.1;§1. Introduction;354
24.2;§2. Total Positivity;356
24.3;§2. Total Positivity;356
24.4;§3. Consequences of Total Positivity;359
24.5;§4. Other Properties;363
24.6;§5. Surfaces;365
24.7;References;369
25;Chapter 19. Geometric Continuity;374
25.1;§1. Introduction;374
25.2;§2. Geometric Arc Length Continuity for Curves;375
25.3;§3. Differential Geometry of Curves;378
25.4;§4. Geometric Frenet Frame Continuity for Curves;380
25.5;§5. Geometric Continuity for Surfaces;382
25.6;§6. Applications;386
25.7;References;388
26;Chapter 20. Curvature Continuous Triangular Interpolants;394
26.1;§1. Introduction;394
26.2;§2. Preliminary Remarks and Statement of the Problem;394
26.3;§3. The Geometric Hermite-operator;395
26.4;§4. A Transfinite VC2-patch;396
26.5;§5. Discretization of the Transfinite VC2-patch;400
26.6;§6. Curvature Estimation for Smooth Surface Design;404
26.7;References;405
27;Chapter 21. Box-Spline Surfaces;406
27.1;§1. Introduction;406
27.2;§2. Definition and Basic Properties;407
27.3;§3. Cardinal Splines;413
27.4;§4. Subdivision Algorithms;415
27.5;§5. Selected Theorems;420
27.6;§6. Bibliographic Comments;422
27.7;References;422
28;Chapter 22. Parallelization of the Subdivision Algorithm for Intersection of Bézier Curves on the FPS T20;424
28.1;§1. Introduction;424
28.2;§2. The FPS T20;424
28.3;3. Parallelization of an Algorithm;425
28.4;§4. The Bézier Algorithm;425
28.5;§5. Parallelization;426
28.6;§6. Vectorization;428
28.7;§7. Experimental Results;429
28.8;§8. Conclusion;431
28.9;References;432
29;Chapter 23. Composite Quadrilateral Finite Elements of Class C;434
29.1;§1. Introduction;434
29.2;§2. FVS Quadrilaterals of Class C;435
29.3;§3. FVS Quadrilaterals of Class C;437
29.4;References;438
29.5;Chapter 4. A Knot Removal Strategy for Scattered Data in R;440
29.6;§1. Introduction;440
29.7;§2. Measure of the Significance of Each Point;441
29.8;§4. Control Points From One Triangle to Another;443
29.9;§5. On Bell's Approximant;444
29.10;§6. Knot Removal;445
29.11;§7. Remarks and Comments;446
29.12;References;446
30;Chapter 24. Interpolation Systems and the Finite Element Method;448
30.1;§1. Introduction;448
30.2;§2. Notation, Definitions, and Previous Results;449
30.3;§3. Triangular Finite Elements of Class k and Type;450
30.4;§4. Remarks;453
30.5;5. Ck-continuity of Finite Elements of Type;454
30.6;References;454
31;Chapter 25. Uniform Bivariate Hermite Interpolation;456
31.1;§1. Introduction;456
31.2;§2. Interpolations with Few Knots;458
31.3;§3. Uniform Hermite Interpolation with Second Derivatives;459
31.4;§4. The General Case;464
31.5;References;465
32;Chapter 26. A Survey of Applications of an Affine Invariant Norm;466
32.1;§1. Introduction;466
32.2;§2. Applications to Scattered Data Interpolation;470
32.3;§3. Applications to Knot Selection;476
32.4;§4. Applications to Triangulations;478
32.5;Acknowledgments;484
32.6;References;487
33;Chapter 27. An Algorithm for Smooth Interpolation to Scattered Data in R;490
33.1;§1. Notation;490
33.2;§2. Interpolation Problem;491
33.3;§3. Choice of the Degree k;492
33.4;§4. A Representation for Elements of the Space Sm2m+1;494
33.5;§5. Outline of the Algorithm;497
33.6;§6. Numerical Experiments;499
33.7;References;500
34;Chapter 28. Some Remarks on Three B-Spline Constructions;502
34.1;§1. Introduction;502
34.2;§2. Bézier Polynomials;502
34.3;§3. de Boor's Algorithm;503
34.4;§4. Boehm's Construction;504
34.5;§5. A New Development of B-splines;504
34.6;§6. Proving de Boor's Algorithm;506
34.7;References;507
35;Chapter 29. Modified B-Spline Approximation for Quasi-Interpolation or Filtering;510
35.1;§1. Introduction;510
35.2;§2. Transfer Function;511
35.3;§3. Choosing the Coefficients bj;513
35.4;§4. Conclusions;519
35.5;References;519
36;Chapter 30. Design Tools for Shaping Spline Models;520
36.1;§1. Introduction;520
36.2;§2. Surfaces from Curves;521
36.3;§3. Making Solids from Surfaces;533
36.4;§4. Modifying Surface Shapes;535
36.5;§5. Examples;536
36.6;§6. Conclusions;537
36.7;Acknowledgements;538
36.8;References;538
37;Chapter 31. A Process Oriented Design Method for Three-dimensional CAD Systems;542
37.1;§1. Introduction;542
37.2;§2. Process Oriented Solid Modeling;543
37.3;§3. Modeling of Complex Shaped Bodies;545
37.4;§4. Examples;547
37.5;§5. Conclusion;549
37.6;References;549
38;Chapter 32. Open Questions in the Application of Multivariate B-splines;550
38.1;§1. Introduction;550
38.2;§2. Multivariate B-splines;550
38.3;§3. Scattered Data Contouring;551
38.4;§4. Field Analysis;555
38.5;§5. Multivariate B-splines Still Outside the Multivariate Theory;556
38.6;References;557
39;Chapter 33. On Global GC2 Convexity Preserving Interpolation of Planar Curves by Piecewise Bézier Polynomials;560
39.1;§1. Global Parametric Spline Interpolation;560
39.2;§2. Cubic Pieces at Inflection Points;562
39.3;§3. Straight Sections;565
39.4;§4. Not-a-knot Boundary Conditions;566
39.5;References;568
40;Chapter 34. Best Interpolation with Free Nodes by Closed Curves;570
40.1;§1. Introduction;570
40.2;§2. Existence;571
40.3;§3. Uniqueness;575
40.4;§4. Discussion;579
40.5;References;580
41;Chapter 35. Segmentation Operators on Coons' Patches;582
41.1;§1. Introduction;582
41.2;§2. Introduction to the Theory of Coons' Patches;583
41.3;§3. Segmentation Operators on Coons' Patches;587
41.4;§4. Segmentation of Subclasses;589
41.5;§5. Segmentation of Tensor-Product Surfaces;591
41.6;References;593
42;Chapter 36. neral Subdivision Theorem for Bézier Triangles;594
42.1;§1. Introduction;594
42.2;§2. The Blossoming Principle;595
42.3;§3. Subdivision Algorithms for Bézier Triangles;597
42.4;§4. Conclusions;601
42.5;References;601
43;Chapter 37. Cardinal Interpolation with Translates of Shifted Bivariate Box-Splines;604
43.1;§1. Introduction;604
43.2;§2. Exponential Euler Splines;605
43.3;§3. Correctness of Cardinal Interpolation: General Results;606
43.4;§4. Correctness of Cardinal Interpolation: Low Order Splines;607
43.5;§5. Further Remarks;612
43.6;Acknowledgement;612
43.7;References;612
44;Chapter 38. Approximation of Surfaces Constrained by a Differential Equation Using Simplex Splines;614
44.1;§1. Introduction;614
44.2;§2. Bivariate Simplex Splines;615
44.3;§3. Surfaces Constrained by a Differential Equation;617
44.4;§4. Inner Products of Lowest Order Splines;618
44.5;§5. Evaluation of a Model Problem;618
44.6;References;620
45;Chapter 39. A Construction for VC1 Continuity of Rational Bézier Patches;622
45.1;§1. Introduction;622
45.2;§2. Tangential Directions for Rectangular Patches;623
45.3;§3. Tangential Directions for Triangular Patches;624
45.4;§4. The VC1 Construction;626
45.5;§5. Farin's Transition Principle;630
45.6;§6. Conclusions;631
46;References;631