Laurent / Le Méhauté / Schumaker Curves and Surfaces

1. Auflage 2014
ISBN: 978-1-4832-6387-8
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 532 Seiten, Web PDF

Curves and Surfaces provides information pertinent to the fundamental aspects of approximation theory with emphasis on approximation of images, surface compression, wavelets, and tomography. This book covers a variety of topics, including error estimates for multiquadratic interpolation, spline manifolds, and vector spline approximation. Organized into 77 chapters, this book begins with an overview of the method, based on a local Taylor expansion of the final curve, for computing the parameter values. This text then presents a vector approximation based on general spline function theory. Other chapters consider a nonparametric technique for estimating under random censorship the amplitude of a change point in change point hazard models. This book discusses as well the algorithm for ray tracing rational parametric surfaces based on inversion and implicitization. The final chapter deals with the results concerning the norm of the interpolation operator and error estimates for a square domain. This book is a valuable resource for mathematicians.

Autoren/Hrsg.

Laurent, Pierre-Jean

Le Méhauté, Alain

Schumaker, Larry L.

Inhaltsverzeichnis

1;Front Cover;1
2;Curves and Surfaces;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE;11
6;CONTRIBUTORS;12
7;Chapter 1. Parametrization for Data Approximation;20
7.1;§1. Introduction;20
7.2;§2. Local Taylor Expansion of the Curve;20
7.3;§3. Criteria;21
7.4;§4. Application to Approximation with B-Splines;22
7.5;References;23
8;Chapter 2. A Vector Spline Approximation With Application to Meteorology;24
8.1;§1. Introduction;24
8.2;§2. The Minimization Problem;25
8.3;§3. Solution of P
a,ß;26
8.4;§4. Limit Problems;28
8.5;§5. Numerical Results;28
8.6;References;29
9;Chapter 3. Kernel Estimation in Change-Point Hazard Rate Models;30
9.1;§1. Introduction;30
9.2;§2. The Model;31
9.3;§3. Main Results;33
9.4;§4. Simulation Results and Concluding Remarks;34
9.5;References;35
10;Chapter 4. Spline Manifolds;36
10.1;§1. Introduction;36
10.2;§2. Basic
Tools;38
10.3;§3. A Fundamental Result;40
10.4;§4. Spline Manifolds;41
10.5;§5. P(D)-positive Vectorial Distributions;43
10.6;§6. Extension of P(D)-spline;44
10.7;References;44
11;Chapter 5. Use of Simulated Annealing to Construct Triangular Facet Surfaces;46
11.1;§1. Introduction;46
11.2;§2. Optimal Triangulations;47
11.3;§3. Locally Optimal Triangulations and Edge Swapping;47
11.4;§4. Simulated Annealing;48
11.5;§5. An Example;49
11.6;§6. Conclusions;51
11.7;References;51
12;Chapter 6.
G1 and G2 Continuity Between (SBR) Surfaces;52
12.1;§1. Geometric Framework;52
12.2;§2. Analytical Conditions;53
12.3;§3. Geometric Continuity Between Two Rectangular (SBR);53
12.4;References;55
13;Chapter 7. Ray Tracing Rational Parametric Surfaces;56
13.1;§1. Introduction;56
13.2;§2. Implicitization;57
13.3;§3. Intersection Problem;59
13.4;§4. Discussion;61
13.5;References;61
14;Chapter 8. Energy-Based Segmentation of Sparse Range Data;62
14.1;Abstract;62
14.2;§1. Segmentation: Introduction and Background;62
14.3;§2. Definition of the Model of World Surfaces;63
14.4;§3. Experimentation;66
14.5;References;69
15;Chapter 9. Error Estimates for Multiquadric Interpolation;70
15.1;§1. Introduction and Statement of Main Result;70
15.2;§2. Proof of Theorem 1;72
15.3;References;77
16;Chapter 10. A Geometrical Analysis for a Data Compression of 3D Anatomical Structures;78
16.1;§1. Introduction;78
16.2;§2. Data Format;78
16.3;§3. Feature Extraction;79
16.4;§4. Data Compression;81
16.5;§5. Contour Reconstruction;82
16.6;§6. Conclusion;83
16.7;References;84
17;Chapter 11. Ck Continuity of (SBR) Surfaces;86
17.1;§1. Framework;86
17.2;§2. Rectangular (SBR) Surfaces;86
17.3;§3. Triangular (SBR) Surfaces;88
17.4;References;89
18;Chapter 12. A Note on Piecewise Monotonie Bivariate Interpolation;90
18.1;§1. Introduction;90
18.2;§2. Outline of the Algorithm;91
18.3;§3. Conclusions;93
18.4;References;93
19;Chapter 13. Real-Time Signal Analysis with Quasi-Interpolatory Splines and Wavelets;94
19.1;§1. Introduction;94
19.2;§2. Spline Sampling of Digital Signals;95
19.3;§3. Wavelet Signal Decomposition;98
19.4;§4. Wavelet Signal Reconstruction;99
19.5;References;101
20;Chapter 14. Polynomial Expansions for Cardinal Interpolants and Orthonormal Wavelets;102
20.1;§1. Introduction;102
20.2;§2. The Case of Real
F;103
20.3;§3. The Case of Complex
F;105
20.4;§4. Construction of Quasi-Interpolation Operators;108
20.5;References;109
21;Chapter 15. Realtime Pipelined Spline Data Fitting for Sketched Curves;110
21.1;§1. Introduction;110
21.2;§2. Background;111
21.3;§3. Data Reduction;111
21.4;§4. Pipelining the Algorithm;114
21.5;§6. Remarks;120
21.6;References;120
22;Chapter 16. Remarks on Digital Terrain Modelling Accuracy;122
22.1;§1. On Terrain Modelling Accuracy;122
22.2;§2. The Displacement-Buckling Approach;123
22.3;§3. A Graphical Approach;124
22.4;References;125
23;Chapter 17. Convexity and Bernstein-Bézier Polynomials;126
23.1;§1. Introduction;126
23.2;§2. Some Prerequisites;128
23.3;§3. Planar Curves;131
23.4;§4. Convexity of Functional Bézier Surfaces;133
23.5;§5. Convexity of Piecewise Polynomial Surfaces;141
23.6;§6. Convexity of Parametric Bézier Patches;143
23.7;§7. Converse Theorems on Convexity;146
23.8;References;150
24;Chapter 18. How to Draw a Curve Using Geometrical Data;154
24.1;§1. Introduction;154
24.2;§2. Algebraic Regression;155
24.3;§3. Automatic Ordering;156
24.4;References;157
25;Chapter 19. The Generation of an Aerodynamical Propeller Using Partial Differential Equations;158
25.1;§1. Introduction;158
25.2;§2. The PDE Method;158
25.3;§3. Generating the Propeller;159
25.4;§4. Aerodynamics;160
25.5;§5. Influence of Parameters;161
25.6;References;161
26;Chapter 20. Fast Computation of Cross-Validated Robust Splines and Other Non-linear Smoothing Splines;162
26.1;§1. Introduction;162
26.2;§2. Choice of p by
Generalized Cross-validation;164
26.3;§3. Monte-Carlo Computation of Trace Terms;165
26.4;§4. Numerical Experiments;165
26.5;References;167
27;Chapter 21. Szasz-Mirakyan Quasi-interpolants;168
27.1;§1. Introduction and Definitions;168
27.2;§2. Norms of the Left Quasi-interpolants;170
27.3;§3. Woronovskaya-type Relation;174
27.4;Acknowledgements;175
27.5;References;175
28;Chapter 22. Statistical Check On The Smoothing Parameter of a Method for Inversion of Fourier Series;176
28.1;References;179
29;Chapter 23. A General Method of Treating Degenerate Bézier Patches;180
29.1;§1. Introduction;180
29.2;§2. Computing Geometric Features;181
29.3;§3. Geometric Continuity Constraints over Degenerate Patches;182
29.4;References;183
30;Chapter 24. G1 Smooth Connection Between Rectangular and Triangular Bézier Patches at a Common Corner;184
30.1;§1. G1 Continuity Between Two Adjacent Bézier Patches;184
30.2;§2. G1 Continuity Around a Mixed N-patch Corner;185
30.3;References;187
31;Chapter 25. Regularity Conditions for a Class of Geometrically Continuous Curves and Surfaces;188
31.1;§1. Introduction;188
31.2;§2. Regularity Conditions for Bell-Shaped Functions;190
31.3;§3. Two Classes of Bell-Shaped Functions;192
31.4;References;195
32;Chapter 26. Splines and Digital Signal Processing;196
32.1;§1. Introduction;196
32.2;§2. B-Spline Digital Filters;197
32.3;References;199
33;Chapter 27. B-Rational Curves and Surfaces N-Rational Splines;200
33.1;§1. Introduction;200
33.2;§2. General Framework;200
33.3;§3. The (BR) Curves;201
33.4;§4. The N-rational Splines;202
33.5;§5. The (SBR) Surfaces;202
33.6;References;203
34;Chapter 28. Reparametrizations of Polynomial and Rational Curves;204
34.1;§1. Introduction;204
34.2;§2. Homographie Transformation;205
34.3;§3. Rational Quadratic Transformation;205
34.4;§4. Conclusion;207
34.5;References;207
35;Chapter 29. Numerical Stability of Geometric Algorithms;208
35.1;References;211
36;Chapter 30. Solving Implicit ODEs by Simplicial Methods;212
36.1;§1. Introduction;212
36.2;§2. PL Approximation of Implicit Manifolds;212
36.3;§3. Implicit Differential Equations and Singularities of Mappings;213
36.4;§4. A Simplicial Method to Solve Implicit ODEs;214
36.5;References;215
37;Chapter 31. On the Power of a posteriori Error Estimation for Numerical Integration and Function Approximation;216
37.1;§1. Introduction;216
37.2;§2. Proof of the Theorems;220
37.3;References;226
38;Chapter 32. Using the Refinement Equations for the Construction of Pre-Wavelets II: Powers of Two;228
38.1;§1. Introduction;228
38.2;§2. Multiresolution Analysis;230
38.3;§3. Symbol Calculus;238
38.4;§4. Stability;245
38.5;§5. Linear Independence;250
38.6;§6. Pre-Wavelet Decomposition;252
38.7;§7. Wavelet Decomposition;255
38.8;§8. Extensibility;260
38.9;References;263
39;Chapter 33. Elastica and Minimal-Energy Splines;266
39.1;§1. Elastica;266
39.2;§2. Minimal-energy Spline Segments;267
39.3;§3. Minimal-energy Splines;268
39.4;References;269
40;Chapter 34. A Distributed Algorithm for Surface/Plane Intersection;270
40.1;§1. Introduction;270
40.2;§2. The Subdivision Step;271
40.3;§3. The Intersection Step;271
40.4;§4. Experimental Examples;271
40.5;§5. Conclusions;273
40.6;References;273
41;Chapter 35. Construction of Exponential Tension B-splines of Arbitrary Order;274
41.1;§1. B-splines;274
41.2;§2. Conversion to a Bézier-like Form;275
41.3;References;277
42;Chapter 36. On the Almost Sure Limit of Probabilistic Recovery;278
42.1;References;286
43;Chapter 37. A New Curve Tracing Algorithm and Some Applications;288
43.1;§1. Introduction;288
43.2;§2. Curve Tracing Algorithm;289
43.3;§3. Applications;290
43.4;References;291
44;Chapter 38. Pseudo-Cubic Weighted Splines Can Be C2 or G2;292
44.1;§1. Introduction;292
44.2;§2. General Weighted Interpolating Spline;292
44.3;§3. Interpolating q-spline of
Order 2;293
44.4;§4. Smoothing q-spline of
Order 2;294
44.5;§5. Polar Representation of
q-splines;294
44.6;References;295
45;Chapter 39. Composite Cr-Triangular Finite Elements of PS Type on a Three Direction Mesh;296
45.1;§1. Introduction;296
45.2;§2. PS Triangles of Class C2s;297
45.3;§3. PS Triangles of Class C2s+1;298
45.4;References;298
46;Chapter 40. Dynamic Segmentation: Finding the Edge With Snake Splines;300
46.1;§1. Introduction;300
46.2;§2. Modelling Curves Or Surfaces With Spline Functions;301
46.3;§3. Strength Fields and Potential Convolving;302
46.4;§4. Adaptative Evolution of the Snake-Spline;303
46.5;§5. Preliminary Results;304
46.6;§6. Conclusion;304
46.7;References;304
47;Chapter 41. Recent Developments in the Strang-Fix Theory for Approximation Orders;306
47.1;§1. Introduction;306
47.2;§2. The Strang-Fix Theory;308
47.3;§3. Functions with Rapid Decay;309
47.4;References;313
48;Chapter 42. Aligning Frames with the Tangent Curve of a B-spline Curve;314
48.1;§1. Introduction;314
48.2;§2. Linear Interpolation of Frames in 2D;315
48.3;§3. 5-spline Approximation of Frames in 2D;316
48.4;§4. Frame Splines in 3D;317
48.5;References;317
49;Chapter 43. Error Estimates for Interpolation by Generalized Splines;318
49.1;§1. Introduction;318
49.2;§2. Variational Formulation;319
49.3;§3. Error Estimates;322
49.4;§4. Miscellaneous Examples and Remarks;325
49.5;References;326
50;Chapter 44. Varying the Shape Parameters of Rational Continuity;328
50.1;§1. Introduction;328
50.2;§2. Rational Continuity;329
50.3;§3. Basis Functions;330
50.4;§4. Conclusion;333
50.5;References;333
51;Chapter 45. Detecting Cusps and Inflection Points in Curves;336
51.1;§1. Introduction;336
51.2;§2. Parametric Curves;337
51.3;§3. Necessary Condition for Cusps;337
51.4;§4. Necessary and Sufficient Condition for Cusps;338
51.5;§5. Proper Parametrizations;339
51.6;References;340
52;Chapter 46. Image-like Surfaces: Parallel Least Squares Approximation Methods;342
52.1;§1. Introduction;342
52.2;§2. Parallel Least Squares Methods;343
52.3;References;345
53;Chapter 47. Local Kriging Interpolation: Application to Scattered Data on the Sphere;346
53.1;§1. Introduction;346
53.2;§2. Local Kriging Interpolation or Approximation on the Sphere;346
53.3;§3. Conclusion;349
53.4;References;349
54;Chapter 48. Best Approximation of Circle Segments by Quadratic Bézier Curves;352
54.1;§1. Introduction;352
54.2;§2. Interpolatory Approximation;353
54.3;§3. General Quadratic Approximation;354
54.4;§4. Numerical Comparisons;357
54.5;Acknowledgements;357
54.6;References;357
55;Chapter 49. A Procedure for Determining Starting Points for a Surface/Surface Intersection Algorithm;358
55.1;§1. Introduction;358
55.2;§2. The Algorithm in Connection with the SSI Algorithm;359
55.3;§3. Correctness;360
55.4;§4. Efficiency;361
55.5;References;361
56;Chapter 50. Norms of Inverses for Matrices Associated with Scattered Data;362
56.1;§1. Introduction;362
56.2;References;368
57;Chapter 51. 2D Sampling in Tomography;370
57.1;§1. Introduction;370
57.2;§2. The Radon Inversion Formula;371
57.3;§3. Shannon's Sampling Theorem;372
57.4;§4. The Sampling Theorem of Petersen-Middleton;373
57.5;§5. The Sampling Theorem of Beurling;375
57.6;§6. Fourier Reconstruction;376
57.7;References;377
58;Chapter 52. Subdividing Multivariate Polynomials Over Simplices in Bernstein-Bézier Form Without de Casteljau Algorithm;380
58.1;§1. Introduction;380
58.2;§2. Three Alternatives for Subdividing Bernstein Polynomials;381
58.3;§3. Remarks;383
58.4;References;383
59;Chapter 53. Geometrically Smooth Interpolation by Triangular Bernstein-Bézier Patches With Coalescent Control Points;384
59.1;§1. Description of the Problem;384
59.2;§2. Degenerate Polynomial Patches;386
59.3;References;387
60;Chapter 54. Curve Fitting Using NURBS;388
60.1;§1. Introduction;388
60.2;§2. Fitting a Parametric Quadratic B-spline Curve;388
60.3;§3. Searching Conies;389
60.4;§4. NURBS Representation of Conies;390
60.5;§5. NURBS Representation of the Whole Curve;391
60.6;References;391
61;Chapter 55. Univariate Multiquadric Interpolation: Some Recent Results;392
61.1;§1. Introduction;392
61.2;§2. Conditions for O(h) Accuracy;395
61.3;§3. Conditions for
O(h2) Accuracy;399
61.4;§4. A Property of the
Multiquadric Parameter c;400
61.5;§5. Discussion;402
61.6;References;402
62;Chapter 56. Periodic Spline Interpolation of Functions of Bounded Variation;404
62.1;§1. Introduction;404
62.2;§2. Statement of Results;405
62.3;§3. Proofs;406
62.4;References;407
63;Chapter 57. Least Squares Fitting by Linear Splines on Data Dependent Triangulations;408
63.1;§1. Introduction;408
63.2;§2. Choosing the Vertex Set;409
63.3;§3. Estimating the Vertex Values;409
63.4;§4. Data Dependent Triangulations;409
63.5;§5. Remarks;411
63.6;References;411
64;Chapter 58. How to Build Quasi-Interpolants: Application to Polyharmonic B-Splines;412
64.1;Introduction;412
64.2;§1. "Elementary" Quasi-interpolants;414
64.3;§2. "High Level Quasi-interpolants";418
64.4;§3. Conclusion;422
64.5;References;423
65;Chapter 59. Algorithms for Local Convexity of Bézier Curves and Surfaces;424
65.1;§1. Introduction;424
65.2;§2. Bézier Curves;424
65.3;§3. Bézier Surfaces;426
65.4;§4. Conclusions;427
65.5;References;427
66;Chapter 60. Polynomial N-sided Patches;428
66.1;§1. Introduction;428
66.2;§2. Tangent Plane Continuity Between Two Patches;429
66.3;§3. Twist Compatibility;429
66.4;§4. Construction of Cross Derivatives;430
66.5;§5. Patch Equation;431
66.6;References;431
67;Chapter 61. Cubic Recursive Division With Bounded Curvature;432
67.1;§1. Introduction;432
67.2;§2. Definitions;433
67.3;§3. Cubic Constructions;434
67.4;§4. A New Construction;435
67.5;§5. Properties;435
67.6;§6. Limitations;435
67.7;References;435
68;Chapter 62. .-Convergence, A Criterion for Linear Approximation;436
68.1;§1. Definitions;436
68.2;§2. The Principle;437
68.3;§3. B-spline Approximation;437
68.4;§4. Tri-quadratic Interpolation;440
69;Chapter 63. Bernstein-Type Quasi-Interpolants;442
69.1;§1. Introduction;442
69.2;§2. Construction of Operators;443
69.3;§3. Some Applications;445
69.4;References;447
70;Chapter 64. Extension of the Problem of Best Interpolating Parametric Curves to L-Splines;448
70.1;§1. Introduction;448
70.2;§2. Existence;449
70.3;§3. Characterization;451
70.4;§4. Uniqueness;452
70.5;References;453
71;Chapter 65. Adaptive G1 Approximation of Range Data Using Triangular Patches;454
71.1;§1. Introduction;454
71.2;§2. Adaptive Triangulation;454
71.3;§3. A Triangular Gregory-Bézier Patch Model;456
71.4;§4. Adaptive G1 Approximation using Triangular Patches;456
71.5;References;457
72;Chapter 66. Universal Splines and Geometric Continuity;458
72.1;§1. Introduction;458
72.2;§2. Geometrie Continuity;459
72.3;§3. Universal Splines;460
72.4;§4. Constructing the Spline Control Points;462
72.5;§5. Constructing the Bézier Points;464
72.6;§6. Conclusion;465
72.7;References;465
73;Chapter 67. Procedural Construction of Patch-Boundary Curves;466
73.1;§1. The Construction Process;466
73.2;§2. Choice of Curve Parameters;467
73.3;§3. Opposite Edge Method;467
73.4;§4. Filling in Patches and Shape Parameters;469
73.5;References;469
74;Chapter 68. Efficient Computation of Multiple Knots Nonuniform Spline Functions;470
74.1;§1. Introduction;470
74.2;§2. The Derivative/summation Approach;471
74.3;§3. Generation of Q(u) from
Q;472
74.4;References;473
75;Chapter 69. Chebyshev Approximation in IRn by
Curves and Linear Manifolds;474
75.1;§1. Introduction;474
75.2;§2. Approximation by a Parameterized Curve;475
75.3;§3. Approximation by a Straight Line;475
75.4;§4. Approximation by Linear Manifolds;476
75.5;References;476
76;Chapter 70. A Building Method for Hierarchical Covering Spheres of a Given Set of Points;478
76.1;§1. The Covering Spheres;478
76.2;§2. Building the Spheres;479
76.3;§3. Uses;480
76.4;References;481
77;Chapter 71. The Variational Approach to Shape Preservation;482
77.1;§1. Introduction;482
77.2;§2. The Norm Minimizing Constrained Splines;483
77.3;§3. The Semi-Norm Case;488
77.4;§4. Numerical Algorithms;492
77.5;§5. Best Constrained Interpolation in Banach Spaces;493
77.6;References;495
78;Chapter 72. Spline Fitting Numerous Noisy Data With Discontinuities;498
78.1;§1. Introduction;498
78.2;§2. B-spline Fitting;499
78.3;§3. Knots Determination;500
78.4;§4. Discussion;501
78.5;References;501
79;Chapter 73. B-Spline Surfaces for Real-time Shape Design;502
79.1;§1. Introduction;502
79.2;§2. B-spline Surface Generation;503
79.3;§3. Implementation and Further Research;505
79.4;References;505
80;Chapter 74. Conversion of a Composite Trimmed Bézier Surface Into Composite Bézier Surfaces;506
80.1;§1. Introduction;506
80.2;§2. Definition of the Problem;506
80.3;§3. Conditions for a Solution;507
80.4;§4. The Algorithm;508
80.5;§5. Unsolved Problems;509
80.6;§6. Conclusions;510
80.7;References;510
81;Chapter 75. Multivariate Model Building With Additive Interaction and Tensor Product Thin Plate Splines;512
81.1;§1. Introduction;512
81.2;§2. Multiple Smoothing Parameters, Splines Based on W2m;514
81.3;§3. Thin Plate Splines;516
81.4;§4. Grand Models, Model Selection;518
81.5;§5. An Example;519
81.6;§6. Diagnostics;519
81.7;§7. Post Analysis Diagnostics, or Accuracy Statements;521
81.8;References;524
82;Chapter 76. Recursion Relations for 4 x 4 Determinants Related to Rational Cubic Bézier Curves;526
82.1;§1. Introduction;526
82.2;§2. The 4 x 4 Determinant Method;527
82.3;§3. The 4 x 4 Determinants before and after Subdivision;529
82.4;§4. Conclusion;531
82.5;References;531
83;Chapter 77. Lagrange Interpolation by Quadratic Splines On a Quadrilateral Domain of
IR2;532
83.1;§1. Introduction;532
83.2;§2. Notations and Problem;532
83.3;§3. Interpolation Operator and Error Estimates for a Square;533
83.4;Acknowledgements;535
83.5;References;535