Lyche / Schumaker Mathematical Methods in Computer Aided Geometric Design II

1. Auflage 2014
ISBN: 978-1-4832-5798-3
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 644 Seiten, Web PDF

Mathematical Methods in Computer Aided Geometric Design II covers the proceedings of the 1991 International Conference on Curves, Surfaces, CAGD, and Image Processing, held at Biri, Norway. This book contains 48 chapters that include the topics of blossoming, cyclides, data fitting and interpolation, and finding intersections of curves and surfaces. Considerable chapters explore the geometric continuity, geometrical optics, image and signal processing, and modeling of geological structures. The remaining chapters discuss the principles of multiresolution analysis, NURBS, offsets, radial basis functions, rational splines, robotics, spline and Bézier methods for curve and surface modeling, subdivision, terrain modeling, and wavelets. This book will prove useful to mathematicians, computer scientists, and advance mathematics students.

Autoren/Hrsg.

Lyche, Tom

Schumaker, Larry L.

Inhaltsverzeichnis

1;Front Cover;1
2;Mathematical Methods in Computer Aided Geometric Design II;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE;9
6;PARTICIPANTS;10
7;Chapter 1. Symmetrizing Multiaffine Polynomials;20
7.1;§1. Introduction and Motivation;20
7.2;§2. Cubics;22
7.3;§3. Quartics, Quintics, and Sextics;23
7.4;§4. Observations on Conversion to B-spline Form;25
7.5;§5. Open Questions;25
7.6;References;26
8;Chapter 2. Norm Estimates for Inverses of Distance Matrices;28
8.1;§1. Introduction;28
8.2;§2. The Univariate Case for the Euclidean Norm;29
8.3;§3. The Multivariate Case for the Euclidean Norm;32
8.4;§4. Fourier Transforms and Bessel Transforms;33
8.5;§5. The Least Upper Bound for Subsets of the Integer Grid;36
8.6;References;36
9;Chapter 3. Numerical Treatment of
Surface–Surface Intersection and Contouring;38
9.1;§1. Introduction;38
9.2;§2. Lattice Evaluation
(2D Grid–Methods);40
9.3;§3. Marching Based on Davidenko's Differential Equation;42
9.4;§4. Marching Based on Taylor Expansion;45
9.5;§5. Conclusion and Future Extensions;46
9.6;References;47
10;Chapter 4. Modeling Closed Surfaces:
A Comparison of Existing Methods;48
10.1;§1. Introduction;48
10.2;§2. Subdivision Schemes;49
10.3;§3. Discrete Interpolation;51
10.4;§4. Algebraic Interpolation;51
10.5;§5. Transfinite
Interpolation;52
10.6;§6. Octree and Face Octree Representations;52
10.7;§7. Discussion of These Modeling Schemes;54
10.8;References;59
11;Chapter 5. A New Characterization of Plane
Elastica;62
11.1;§1. Introduction;62
11.2;§2. A Characterization of Elástica by their Curvature Function;63
11.3;§3. A Characterizing Representation Theorem;72
11.4;References;75
12;Chapter 6. POLynomials, POLar Forms, and InterPOLation;76
12.1;§1. Introduction;76
12.2;§2. Algebraic Definition of Polar Curves;76
12.3;§3. Interpolation;81
12.4;§4. Conclusion and a Few Historical Remarks;85
13;Chapter 7. Pyramid Patches Provide
Potential Polynomial Paradigms;88
13.1;§1. Introduction;88
13.2;§2. Linear Independence of Families of Lineal Polynomials;90
13.3;§3. B-patches for Hn(IRs);97
13.4;§4. Other Pyramid Schemes;100
13.5;§5. B-patches for IIn
(IRs);107
13.6;§6. Degree Raising, Conversion and Subdivision for B-patches;111
13.7;References;119
14;Chapter 8. Implicitizing Rational Surfaces with Base Points by Applying Perturbations and the
Factors of Zero Theorem;120
14.1;§1. Introduction;120
14.2;§2. Mathematical Preliminaries;121
14.3;§3. The Factors of Zero Theorem;123
14.4;§4. Implicitization with Base Points Using the Dixon Resultant;125
14.5;§5. An Implicitization Example;127
14.6;§6. Conclusion and Open Problems;128
14.7;References;128
15;Chapter 9. Wavelets and Multiscale Interpolation;130
15.1;§1. Introduction;130
15.2;§2. Wavelets and Multiresolution
Analysis;132
15.3;§3. Fundamental Scaling Functions;136
15.4;§4. Symmetric and Compactly Supported Scaling Functions;140
15.5;§5. Subdivision Schemes;143
15.6;§6. Regularity;146
15.7;References;151
16;Chapter 10. Decomposition of Splines;154
16.1;§1. Introduction;154
16.2;§2. Decomposition;155
16.3;§3. Decomposing Splines;159
16.4;§4. Box Spline Decomposition;169
16.5;§5. Data Reduction by Decomposition;171
16.6;References;178
17;Chapter 11. A Curve Intersection Algorithm with Processing of Singular Cases: Introduction
of a CHpping Technique;180
17.1;§1. Introduction;180
17.2;§2. Clipping;181
17.3;§3. Singular Cases;183
17.4;§4. Examples;185
17.5;§5. Extension to Surfaces;187
17.6;§6. Conclusion;188
17.7;References;188
18;Chapter 12. Best Approximations of Parametric
Curves by Splines;190
18.1;§1. Introduction;190
18.2;§2. Distances in the Set of Plane Parametric Curves;191
18.3;§3. Non-linear Approximation Theory for Bézier Curves;196
18.4;§4. Numerical Methods and Examples;199
18.5;References;202
19;Chapter 13. An Approximately Cubic Surface Interpolant;204
19.1;§1. Introduction;204
19.2;§2. Notation and Background;205
19.3;§3. Motivation;206
19.4;§4. Approximate Continuity;206
19.5;§5. Geometric Hermite Curves;207
19.6;§6. The Cubic Interpolant;207
19.7;§7. A Piecewise Cubic Surface Scheme;210
19.8;§8. Results;211
19.9;§9. Conclusions;213
19.10;§10. Future Work;213
19.11;References;214
20;Chapter 14. On the Continuity of
Piecewise Parametric Surfaces;216
20.1;§1. Introduction;216
20.2;§2. Characterization of G2 Continuity;217
20.3;§3. G2 Continuity Between Two Surface Patches;217
20.4;§4. G2 Continuity Around an N-patch Corner;218
20.5;§5. Piecewise Representation of G2
Continuous Surfaces;225
20.6;References;225
21;Chapter 15. Stationary and Non-Stationary
Binary Subdivision Schemes;228
21.1;§1. Introduction;228
21.2;§2. Analysis of Stationary Schemes;231
21.3;§3. Analysis of Non-stationary Schemes;232
21.4;References;235
22;Chapter 16.
MQ–Curves are Curves in Tension;236
22.1;§1. Introduction;236
22.2;§2. Properties of General
MQ–Curves;237
22.3;§3. Interpolation;239
22.4;§4. Local Basis Function;241
22.5;§5. Spline Interpolation;243
22.6;References;246
23;Chapter 17. Offset Approximation Improvement
by Control Point Perturbation;248
23.1;§1. Introduction;248
23.2;§2. Background;249
23.3;§3. Getting a Better Approximation of Offsets;250
23.4;§4. Examples;252
23.5;References;256
24;Chapter 18. Curves and Surfaces in Geometrical Optics;258
24.1;§1. Introduction;258
24.2;§2. Envelopes, Offset
Curves, Evolutes and Involutes;261
24.3;§3. Simple Refraction/reflection
Problems;267
24.4;§4. Concluding Remarks;277
24.5;References;278
25;Chapter 19. Evaluation and Properties of
the Derivative of a NURBS Curve;280
25.1;§1. Introduction;280
25.2;§2. Definitions;283
25.3;§3. The Derivative of a Rational B-spline Curve;286
25.4;§4. The Hodograph Property;290
25.5;§5. Bounds on the Derivative;292
25.6;References;293
26;Chapter 20. Hybrid Cubic Bézier Triangle Patches;294
26.1;§1. Introduction;294
26.2;§2. Hybrid Cubic Bezier
Triangle;295
26.3;§3. Non-parametric Hybrid Patches;298
26.4;§4. Cross Boundary Derivatives;300
26.5;§5. Concluding Remarks;304
26.6;References;304
27;Chapter 21. Modelling Geological Structures Using Splines;306
27.1;§1. Introduction;306
27.2;§2. Building Geological Models;307
27.3;§3. Program Structures;313
27.4;References;314
28;Chapter 22. Wonderful Triangle: A Simple, Unified, Algorithmic Approach to Change of Basis Procedures in
Computer Aided Geometric Design;316
28.1;§1. Introduction;316
28.2;§2. Progressive and Pólya Bases;317
28.3;§3. Dual Functionals;321
28.4;§4. Wonderful Triangle;322
28.5;§5. Algorithms — Knot Insertion;325
28.6;§6. Principles of Duality;330
28.7;§7. More Algorithms — Knot Deletion;332
28.8;§8. Still More Algorithms — Factoring the Transformation;335
28.9;§9. Summary, Conclusions, and Future Work;336
28.10;References;337
29;Chapter 23. An Arbitrary Mesh Network Scheme
Using Rational SpUnes;340
29.1;§1. Introduction;340
29.2;§2. The Mesh Network;341
29.3;§3. The Rational Spline Strip Functions;342
29.4;§4. Blending the Strip Functions on Polygonal Domains;344
29.5;§5. Examples and Concluding Remarks;347
29.6;References;348
30;Chapter 24. Bézier Curves and Surface Patches on Quadrics;350
30.1;§1. Introduction;350
30.2;§2. A Suitable Map;351
30.3;§3. Bezier Curves on Quadrics;352
30.4;§4. Bezier Patches on Quadrics;354
30.5;§5. General Solution;357
30.6;§6. Conclusion;360
30.7;References;360
31;Chapter 25. Monotonicity Preserving Interpolation
Using Rational Cubic Bezier Curves;362
31.1;§1. Introduction;362
31.2;§2. Rational Cubic Bézier Representation in an Interval;363
31.3;§3. The Monotonicity Conditions;364
31.4;§4. Determination of the Weights and the First Derivatives;365
31.5;§5. Modification of the Curve;366
31.6;§6. Numerical Examples;367
31.7;§7. Conclusion;369
31.8;References;369
32;Chapter 26. Minimization of Interpolating Spline Curves
with Bounded Derivatives;370
32.1;§1. The Problem;370
32.2;§2. Minimization on the Unit Interval;371
32.3;§3. Examples;373
32.4;References;377
33;Chapter 27. On Piecewise Quadratic
Approximation and Interpolation;382
33.1;§1. Introduction;382
33.2;§2. Placing the Breakpoints;383
33.3;§3. Examples;387
33.4;§4. Interpolation;388
33.5;§5. More Examples;389
33.6;References;389
34;Chapter 28. Non-affine Blossoms
and Subdivision for Q-Splines;390
34.1;§1. Introduction;390
34.2;§2. Polynomials of Degree 4 Satisfying a
q-relation;392
34.3;§3. Q-spline Curves;396
34.4;References;402
35;Chapter 29. On a Class of Data Parametrizations:
Variations on a Theme of Epstein, II;404
35.1;§1. Introduction;404
35.2;§2. Corner Prohibiting Parametrizations;405
35.3;§3. Proof of
the Theorem;407
35.4;§4. Uniform
Parametrization of Nearly Evenly Spaced Data;411
35.5;References;413
36;Chapter 30. Wavelets and Image Compression;414
36.1;§1. Introduction;414
36.2;§2. Wavelet Decomposition of Images: The Haar Transform;415
36.3;§3. Image Compression in
L2(O);417
36.4;§4. Error, Smoothness, and Quantization;418
36.5;§5. Compression in
Lp(O), 1 < . < 8;420
36.6;§6. Examples;423
36.7;References;423
37;Chapter 31. Lower Bounds on the Dimension of Bivariate
Spline Spaces and Generic Triangulations;424
37.1;§1. Introduction;424
37.2;§2. The Main Result;425
37.3;§3. The Lower Bound;426
37.4;§4. Generic Triangulations;431
37.5;References;434
38;Chapter 32. Geometrie Contact of Order .
Between Two Surfaces;436
38.1;§1. Introduction;436
38.2;§2. Frenet-contact of Order .;436
38.3;§3. Comparison Between Fp-contact and
Gp-contact;440
38.4;References;440
39;Chapter 33. On Non-Parametric Constrained Interpolation;442
39.1;§1. Introduction;442
39.2;§2. Straight Line as Constraint Curve;444
39.3;§3. Quadratic Curve as Constraint Curve;445
39.4;§5. Numerical Results;450
39.5;§6. Conclusions;452
39.6;References;453
40;Chapter 34. Tensor Product Slices;454
40.1;§1. Introduction;454
40.2;§2. Tensor Products and Slices;456
40.3;§3. Generally Inherited Properties;458
40.4;§4. Dimension Raising;458
40.5;§5. Ordering of Affine Recurrences;458
40.6;§6. Conversion to Bézier Assimilated Form via Blossoming;459
40.7;§7. Joining a Slice Patch to an Existing Surface;461
40.8;§8. Progress;462
40.9;References;463
41;Chapter 35. Construction of Smooth Surfaces
by Piecewise Tensor Product Polynomials;464
41.1;§1. Introduction;464
41.2;§2. Simple Quadrilateral Meshes;465
41.3;§3. Continuity Conditions;465
41.4;§4. SQM upon Subdivision of a Cube;466
41.5;§5. Further Subdivision of A Cube;470
41.6;References;478
42;Chapter 36. The Virtues of Cyclides in CAGD;480
42.1;§1. Surface Representations;480
42.2;§2. The Dupin Cyclides;483
42.3;§3. Cyclides in Solid Modelling;488
42.4;§4. Summary and Conclusions;494
42.5;References;494
43;Chapter 37. Simple Surfaces Have No Simple
C1 Parametrization;498
43.1;§1. Introduction;498
43.2;§2. Simple Closed Surfaces;499
43.3;§3. Simple Open Surfaces;500
43.4;§4. Concluding Remarks;503
43.5;References;503
44;Chapter 38. Some Tools for Quasi-Interpolation
on Cardinal Grids;504
44.1;Introduction;504
44.2;§1. What is a Quasi-interpolant?;505
44.3;§2. B-approximation Viewed as a Digital Filter;509
44.4;§3. Properties;511
44.5;§4. Modified Approximants;513
44.6;§5. Examples;516
44.7;Conclusion;518
44.8;References;519
45;Chapter 39. Discrete Bézier Curves and Surfaces;520
45.1;§1. Introduction;520
45.2;§2. Discrete Bernstein Basis;521
45.3;§3. Algorithms for Discrete Bezier Curves;522
45.4;§4. Subdivision Algorithm;524
45.5;§5. Extension Algorithms;527
45.6;§6. Joining two Discrete Bézier Curves;528
45.7;§7. Rectangular Discrete Bézier Surfaces;532
45.8;§s. Triangular Discrete Bezier Surfaces;535
45.9;§9. Discrete Bernstein Quasi-interpolants;535
45.10;§10. Dual Basis, Condition Number, Finite Differences;536
45.11;References;537
46;Chapter 40. Rational Geometrie Curve Interpolation;540
46.1;§1. Introduction;540
46.2;§2. Two-point GC2
Hermite Interpolation;543
46.3;§3. Two-point GC1
Hermite Interpolation;548
46.4;§4. Lagrange Interpolation via Hermite Interpolation;550
46.5;§5. Direct Lagrange Interpolation in IR3;550
46.6;§6. Direct Lagrange Interpolation in IR2;551
46.7;§7. Examples;553
46.8;References;557
47;Chapter 41. Curvature Properties of Parametric
Triangular Bézier Patches;560
47.1;§1. Introduction;560
47.2;§2. The Criterion for a Vertex;561
47.3;§3. The Global Criterion;564
47.4;§4. Classification of Polynomial Degree Two Surfaces;566
47.5;References;571
48;Chapter 42. Offsets of Polynomial Bezier Curves:
Hermite Approximation with Error Bounds;572
48.1;§1. Introduction;572
48.2;§2. Interval Bezier Curves;573
48.3;§3. Hermite Approximation of Offset Curves;575
48.4;§4. Polynomial Approximation of Rational Bezier Curves;577
48.5;§5. Numerical Example;579
48.6;§6. Discussion;580
48.7;References;581
49;Chapter 43. Representing Piecewise Polynomials as Linear
Combinations of Multivariate B-Splines;582
49.1;§1. Introduction;582
49.2;§2. Polynomials and Polar Forms;583
49.3;§3. The New B-spline Scheme;585
49.4;§4. Piecewise Polynomials as Linear Combinations of B-splines;587
49.5;References;589
50;Chapter 44. An Explicit Derivation of Discretely Shaped
Beta-spline Basis Functions of Arbitrary Order;590
50.1;§1. Introduction;590
50.2;§2. Geometric Continuity and Beta-constraints;592
50.3;§3. Beta-splines;595
50.4;§4. Beta-constraints Revisited;597
50.5;§5. Solving for
the Beta-polynomials;600
50.6;§6. Conclusion;605
50.7;References;606
51;Chapter 45. Discrete Convolution Schemes;608
51.1;§1. Introduction;608
51.2;§2. Discrete Convolution;609
51.3;§3. Bernstein Basis as Discrete Convolution;610
51.4;§4. Convolution Basis Functions;610
51.5;§5. Basic Properties of the Convolution Basis Functions;612
51.6;§6. Identities for the Convolution Basis Functions;614
51.7;§7. Marsden's Identity and Convolution;614
51.8;§8. Relation to the Discrete B-splines;616
51.9;§9. Dual Functionals;616
51.10;§10. Convolution Curves;616
51.11;§11. Summary;618
51.12;References;618
52;Chapter 46. A Method for Removing the Singularities
from Gregory Surfaces;620
52.1;§1. Introduction;620
52.2;§2. Gregory' s Square;621
52.3;§3. Gregory Surfaces;622
52.4;§4. Gregory Functions as Rational Bezier Functions;623
52.5;§5. Conversion of Gregory Surfaces to Rational Bézier Surfaces;624
52.6;§6. Removal of the Singurality of Gregory Functions;627
52.7;§7. Conclusion;628
52.8;References;628
53;Chapter 47. Bivariate Spline Approximation
by Penalized Least Squares;630
53.1;§1. Introduction;630
53.2;§2. Solution of the Problem;631
53.3;§3. The Algorithm;633
53.4;§4. Numerical Results;634
53.5;References;637
54;Chapter 48. Interpolation with g-splines;638
54.1;§1. Introduction;638
54.2;§2. The g-Spline Space;639
54.3;§3. G1 Constraints;641
54.4;§4. A G1 Interpolation Problem and its Solution;642
54.5;§5. Interpolating Closed Surfaces;647
54.6;References;649