Spline functions are universally recognized as highly effective tools in approximation theory, computer-aided geometric design, image analysis, and numerical analysis. The theory of univariate splines is well known but this text is the first comprehensive treatment of the analogous bivariate theory. A detailed mathematical treatment of polynomial splines on triangulations is outlined, providing a basis for developing practical methods for using splines in numerous application areas. The detailed treatment of the Bernstein-Bézier representation of polynomials will provide a valuable source for researchers and students in CAGD. Chapters on smooth macro-element spaces will allow engineers and scientists using the FEM method to solve partial differential equations numerically with new tools. Workers in the geosciences will find new tools for approximation and data fitting on the sphere. Ideal as a graduate text in approximation theory, and as a source book for courses in computer-aided geometric design or in finite-element methods.
Author(s): Ming-Jun Lai, Larry L. Schumaker
Series: Encyclopedia of Mathematics and its Applications
Edition: 1
Publisher: Cambridge University Press
Year: 2007
Language: English
Pages: 609
Cover......Page 1
About......Page 2
Spline Functions on Triangulations......Page 4
9780521875929......Page 5
Contents......Page 6
Preface......Page 12
1.2. Norms of Polynomials on Triangles ......Page 18
1.3. Derivatives of Polynomials ......Page 19
1.4. Polynomial Approximation in the Maximum Norm ......Page 20
1.5. Averaged Taylor Polynomials ......Page 21
1.6. Polynomial Approximation in the q Norm ......Page 24
1.7. Approximation on Nonconvex ......Page 26
1.8. Interpolation by Bivariate Polynomials ......Page 27
1.9. Remarks ......Page 32
1.10. Historical Notes ......Page 34
2.1. Barycentric Coordinates ......Page 35
2.2. Bernstein Basis Polynomials ......Page 37
2.3. The B-form ......Page 39
2.4. Stability of the B-form Representation ......Page 41
2.5. The deCasteljau Algorithm ......Page 42
2.6. Directional Derivatives ......Page 44
2.7. Derivatives at a Vertex ......Page 48
2.8. Cross Derivatives ......Page 51
2.9. Computing Coefficients by Interpolation ......Page 53
2.10. Conditions for Smooth Joins of Polynomials ......Page 55
2.11. Computing Coefficients From Smoothness ......Page 58
2.12. The Markov Inequality on Triangles ......Page 61
2.13. Integrals and Inner-products of B-polynomials ......Page 62
2.14. Subdivision ......Page 64
2.16. Dual Bases for the Bernstein Basis Polynomials ......Page 66
2.17. A Quasi-interpolant ......Page 68
2.18. The Bernstein Approximation Operator ......Page 69
2.19. Remarks ......Page 74
2.20. Historical Notes ......Page 77
3.1. Control Nets and Control Surfaces ......Page 79
3.3. Positivity of B-patches ......Page 82
3.4. Monotonicity of B-patches ......Page 87
3.5. Convexity of B-patches ......Page 89
3.6. Control Surfaces and Subdivision ......Page 94
3.7. Control Surfaces and Degree Raising ......Page 96
3.8. Rendering a B-patch ......Page 99
3.11. Historical Notes ......Page 101
4.1. Properties of Triangles ......Page 103
4.2. Triangulations ......Page 104
4.4. Euler Relations ......Page 106
4.5. Storing Triangulations ......Page 108
4.6. Constructing Triangulations ......Page 111
4.7. Clusters of Triangles ......Page 113
4.8. Refinements of Triangulations ......Page 114
4.9. Optimal Triangulations ......Page 120
4.10. Maxmin-Angle Triangulations ......Page 121
4.11. Delaunay Triangulations ......Page 126
4.12. Constructing Delaunay Triangulations ......Page 127
4.13. Type-I and Type-II Triangulations ......Page 128
4.14. Quadrangulations ......Page 129
4.15. Triangulated Quadrangulations ......Page 134
4.16. Nested Sequences of Triangulations ......Page 137
4.17. Remarks ......Page 138
4.18. Historical Notes ......Page 141
5.1. The B-form Representation of Splines ......Page 144
5.2. Storing, Evaluating and Rendering Splines ......Page 145
5.3. Control Surfaces and the Shape of Spline Surfaces ......Page 146
5.4. Dimension and a Local Basis for S^0_d(\bigtriangleup)......Page 147
5.5. Spaces of Smooth Splines ......Page 149
5.6. Minimal Determining Sets ......Page 152
5.7. Approximation Power of Spline Spaces ......Page 154
5.8. Stable Local Bases ......Page 158
5.9. Nodal Minimal Determining Sets ......Page 160
5.10. Macro-element Spaces ......Page 163
5.11. Remarks ......Page 164
5.12. Historical Notes ......Page 166
6.1. A C^1 Polynomial Macro-element Space......Page 168
6.2. A C^1 Clough-Tocher Macro-element Space......Page 172
6.3. A C^1 Powell–Sabin Macro-element Space......Page 176
6.4. A C^1 Powell–Sabin-12 Macro-element Space......Page 180
6.5. A C^1 Quadrilateral Macro-element Space......Page 183
6.6. Comparison of C^1 Macro-element Spaces......Page 188
6.7. Remarks ......Page 189
6.8. Historical Notes ......Page 190
7.1. A C^2 Polynomial Macro-element space......Page 191
7.2. A C^2 Clough–Tocher Macro-element Space......Page 195
7.3. A C^2 Powell–Sabin Macro-element Space......Page 199
7.4. A C^2 Wang Macro-element Space......Page 203
7.5. A C^2 Double Clough–Tocher Macro-element......Page 206
7.6. A C^2 Quadrilateral Macro-element Space......Page 209
7.7. Comparison of C^2 Macro-element Spaces......Page 213
7.8. Remarks ......Page 214
7.9. Historical Notes ......Page 215
8.1. Polynomial Macro-element Spaces ......Page 216
8.2. Clough–Tocher Macro-element Spaces ......Page 220
8.3. CT Spaces with Natural Degrees of Freedom ......Page 226
8.4. Powell–Sabin Macro-element Spaces ......Page 231
8.5. PS Spaces with Natural Degrees of Freedom ......Page 237
8.6. Quadrilateral Macro-element Spaces ......Page 243
8.7. Remarks ......Page 248
8.8. Historical Notes ......Page 250
9.1. Dimension of Spline Spaces on Cells ......Page 251
9.2. Dimension of Superspline Spaces on Cells ......Page 255
9.3. Bounds on the Dimension of S^r_d(\bigtriangleup)......Page 257
9.4. Dimension of S^r_d(\bigtriangleup) for d >= 3r + 2......Page 261
9.5. Dimension of Superspline Spaces ......Page 266
9.6. Splines on Type-I and Type-II Triangulations ......Page 270
9.7. Bounds on the Dimension of Superspline Spaces ......Page 272
9.8. Generic Dimension ......Page 279
9.9. The Generic Dimension of S^1_3(\bigtriangleup)......Page 282
9.10. Remarks ......Page 289
9.11. Historical Notes ......Page 291
10.1. Approximation Power ......Page 293
10.3. Approximation Power of S^r_d(\bigtriangleup) for d >= 3r + 2......Page 294
10.4. Approximation Power of S^r_d(\bigtriangleup) for d < 3r + 2......Page 303
10.5. Remarks ......Page 321
10.6. Historical Notes ......Page 323
11.1. Introduction ......Page 325
11.2. Supersplines on Four-cells ......Page 326
11.3. A Lemma on Near-degenerate Edges ......Page 334
11.4. A Stable Local MDS for S^{r,μ}_d(\bigtriangleup)......Page 335
11.5. A Stable MDS for Splines on a Cell ......Page 342
11.6. A Stable Local MDS for S^{r,ρ}_d(\bigtriangleup)......Page 344
11.7. Stability and Local Linear Independence ......Page 345
11.8. Remarks ......Page 348
11.9. Historical Notes ......Page 350
12.1. Type-I Box Splines ......Page 351
12.2. Type-II Box Splines ......Page 360
12.3. Box Spline Series ......Page 364
12.4. The Strang–Fix Conditions ......Page 368
12.5. Polynomial Reproducing Formulae ......Page 372
12.6. Box Spline Quasi-interpolants ......Page 376
12.7. Half Box Splines ......Page 380
12.8. Finite Shift-invariant Spaces ......Page 383
12.9. Remarks ......Page 392
12.10. Historical Notes ......Page 394
13.1. Spherical Polynomials ......Page 395
13.2. Derivatives of Spherical Polynomials ......Page 408
13.3. Spherical Triangulations ......Page 413
13.4. Spaces of Spherical Splines ......Page 414
13.5. Spherical Macro-element Spaces ......Page 423
13.6. Remarks ......Page 424
13.7. Historical Notes ......Page 425
14.2. Projections of Triangulations ......Page 426
14.3. Norms on the Sphere ......Page 431
14.4. Spherical Sobolev Spaces ......Page 433
14.5. Sobolev Seminorms ......Page 436
14.6. Clusters of Spherical Triangles ......Page 438
14.7. Local Approximation by Spherical Polynomials ......Page 440
14.8. The Markov Inequality for Spherical Polynomials ......Page 441
14.9. Spaces with Full Approximation Power ......Page 442
14.10. Remarks ......Page 449
14.11. Historical Notes ......Page 450
15.1. The Space P_d......Page 451
15.2. Barycentric Coordinates ......Page 452
15.3. Bernstein Basis Polynomials ......Page 454
15.4. The B-form of a Trivariate Polynomial ......Page 455
15.5. Stability of the B-form ......Page 457
15.6. The de Casteljau Algorithm ......Page 458
15.7. Directional Derivatives ......Page 459
15.8. B-coefficients and Derivatives at a Vertex ......Page 460
15.9. B-coefficients and Derivatives on Edges ......Page 463
15.10. B-coefficients and Derivatives on Faces ......Page 466
15.11. B-Coefficients and Hermite Interpolation ......Page 468
15.13. Integrals and Inner-products ......Page 469
15.14. Conditions for Smooth Joins ......Page 470
15.15. Approximation Power in the Maximum Norm ......Page 471
15.16. Averaged Taylor Polynomials ......Page 472
15.17. Approximation Power in the q-Norms ......Page 473
15.18. Subdivision ......Page 474
15.20. Remarks ......Page 475
15.21. Historical Notes ......Page 477
16.1. Properties of a Tetrahedron ......Page 478
16.2. General Tetrahedral Partitions ......Page 480
16.3. Regular Tetrahedral Partitions ......Page 481
16.4. Euler Relations ......Page 482
16.5. Constructing and Storing Tetrahedral Partitions ......Page 486
16.6. Clusters of Tetrahedra ......Page 487
16.7. Refinements of Tetrahedral Partitions ......Page 489
16.9. Remarks ......Page 496
16.10 Historical Notes ......Page 497
17.1. C^0 Trivariate Spline Spaces......Page 498
17.2. Spaces of Smooth Splines ......Page 500
17.3. Minimal Determining Sets ......Page 501
17.4. Approximation Power of Trivariate Spline Spaces ......Page 503
17.5. Stable Local Bases ......Page 506
17.6. Nodal Minimal Determining Sets ......Page 507
17.7. Hermite Interpolation ......Page 509
17.8. Dimension of Trivariate Spline Spaces ......Page 511
17.9. Remarks ......Page 516
17.10. Historical Notes ......Page 517
18.1. Introduction ......Page 519
18.2. A C^1 Polynomial Macro-element......Page 520
18.3. A C^1 Macro-element on the Alfeld Split......Page 525
18.4. A C^1 Macro-element on the Worsey–Farin Split......Page 530
18.5. A C^1 Macro-element on the Worsey–Piper Split......Page 534
18.6. A C^2 Polynomial Macro-element......Page 537
18.7. A C^2 Macro-element on the Alfeld Split......Page 541
18.8. A C^2 Macro-element on the Worsey–Farin Split......Page 547
18.9. Another C^2 Worsey–Farin Macro-element......Page 554
18.10. A C^2 Macro-element on the Alfeld-16 Split......Page 561
18.11. A C^r Polynomial Macro-element......Page 565
18.12. Remarks ......Page 574
18.13. Historical Notes ......Page 575
References ......Page 576
Index ......Page 604
