Spline Functions on Triangulations

Spline Functions on Triangulations MING-JUN LAI AND LARRY L. SCHUMAKER CAMBRIDGE UNIVERSITY PRESS

Contents Preface xi Chapter 1. Bivariate Polynomials 1.1. Introduction 1 1.2. Norms of Polynomials on Triangles 1 "1.3. Derivatives of Polynomials 2 1.4. Polynomial Approximation in the Maximum Norm 3 1.5. Averaged Taylor Polynomials 4 1.6. Polynomial Approximation in the q Norm 7 1.7. Approximation on Nonconvex ft 9 1.8. Interpolation by Bivariate Polynomials 10 1.9. Remarks 15 1.10. Historical Notes 17 Chapter 2. Bernstein Bezier Methods for Bivariate Polynomials 2.1. Barycentric Coordinates 18 2.2. Bernstein Basis Polynomials 20 2.3. The B-form 22 2.4. Stability of the B-form Representation 24 2.5. The decasteljau Algorithm 25 2.6. Directional Derivatives 27 2.7. Derivatives at a Vertex 31 2.8. Cross Derivatives 34 2.9. Computing Coefficients by Interpolation 36 2.10. Conditions for Smooth Joins of Polynomials 38 2.11. Computing Coefficients From Smoothness 41 2.12. The Markov Inequality on Triangles -, 44 2.13. Integrals and Inner-products of B-polynomials 45 2.14. Subdivision 47 2.15. Degree Raising 49 2.16. Dual Bases for the Bernstein Basis Polynomials 49 2.17. A Quasi-interpolant 51 2.18. The Bernstein Approximation Operator. 52 2.19. Remarks 57 2.20. Historical Notes 60 Chapter 3. B-Patches 3.1. Control Nets and Control Surfaces 62 3.2. The Convex Hull Property 65 3.3. Positivity of B-patches 65 3.4. Monotonicity of B-patches 70

vi Contents 3.5. Convexity of B-patches 72 3.6. Control Surfaces and Subdivision 77 3.7. Control Surfaces and Degree Raising 79 3.8. Rendering a B-patch 82 3.9. Parametric Patches 84 3.10. Remarks 84 3.11. Historical Notes 84 Chapter 4. Triangulations and Quadrangulations 4.1. Properties of Triangles 86 4.2. Triangulations 87 4.3. Regular Triangulations 89 4.4. Euler Relations 89 4.5. Storing Triangulations 91 4.6. Constructing Triangulations 94 4.7. Clusters of Triangles 96 4.8. Refinements of Triangulations 97 4.9. Optimal Triangulations 103 4.10. Maxmin-Angle Triangulations 104 4.11. Delaunay Triangulations 109 4.12. Constructing Delaunay Triangulations 110 4.13. Type-I and Type-II Triangulations 111 4.14. Quadrangulations 112 4.15. Triangulated Quadrangulations 117 4.16. Nested Sequences of Triangulations 120 4.17. Remarks 121 4.18. Historical Notes. 124 Chapter 5. Bernstein Bezier Methods for Spline Spaces 5.1. The B-form Representation of Splines 127 5.2. Storing, Evaluating and Rendering Splines 128 5.3. Control Surfaces and the Shape of Spline Surfaces 129 5.4. Dimension and a Local Basis for <S (A) 130 5.5. Spaces of Smooth Splines 132 5.6. Minimal Determining Sets 135 5.7. Approximation Power of Spline Spaces 137 5.8. Stable Local Bases. 141 5.9. Nodal Minimal Determining Sets 143 5.10. Macro-element Spaces 146 5.11. Remarks 147 5.12. Historical Notes 149 Chapter 6. C 1 Macro-element Spaces 6.1. A C 1 Polynomial Macro-element Space 151 6.2. A C 1 Clough-Tocher Macro-element Space 155

Contents vii 6.3. A C 1 Powell-Sabin Macro-element Space 159 6.4. A C 1 Powell-Sabin-12 Macro-element Space 163 6.5. A C 1 Quadrilateral Macro-element Space 166 6.6. Comparison of C 1 Macro-element Spaces 171 6.7. Remarks 172 6.8. Historical Notes 173 Chapter 7. C 2 Macro-element Spaces 7.1. AC 2 Polynomial Macro-element space 174 7.2. A C 2 Clough-Tocher Macro-element Space 178 7.3. A C 2 Powell-Sabin Macro-element Space 182 7.4. A C 2 Wang Macro-element Space 186 7.5. A C 2 Double Clough-Tocher Macro-element 189 7.6. A C 2 Quadrilateral Macro-element Space 192 7.7. Comparison of C 2 Macro-element Spaces 196 7.8. Remarks 197 7.9. Historical Notes 198 Chapter 8. C r Macro-element Spaces 8.1. Polynomial Macro-element Spaces 199 8.2. Clough-Tocher Macro-element Spaces 203 8.3. CT Spaces with Natural Degrees of Freedom 209 8.4. Powell-Sabin Macro-element Spaces 214 8.5. PS Spaces with Natural Degrees of Freedom 220 8.6. Quadrilateral Macro-element Spaces 226 8.7. Remarks 231 8.8. Historical Notes. 233 Chapter 9. Dimension of Spline Spaces 9.1. Dimension of Spline Spaces on Cells 234 9.2. Dimension of Superspline Spaces on Cells 238 9.3. Bounds on the Dimension of 5J(A) 240 9.4. Dimension of S r d{/\) for d > 3r + 2 244 9.5. Dimension of Superspline Spaces 249 9.6. Splines on Type-I and Type-II Triangulations 253 9.7. Bounds on the Dimension of Superspline Spaces 255 9.8. Generic Dimension 262 9.9. The Generic Dimension of &} (A) 265 9.10. Remarks 272 9.11. Historical Notes 274 Chapter 10. Approximation Power of Spline Spaces 10.1. Approximation Power 276 10.2. C Splines and Piecewise Polynomials 277 10.3. Approximation Power of <S (A) for d > 3r + 2 277

viii Contents 10.4. Approximation Power of <SJ(A) for d < 3r + 2 286 10.5. Remarks 304 10.6. Historical Notes 306 Chapter 11. Stable Local Minimal Determining Sets 11.1. Introduction 308 11.2. Supersplines on Four-cells 309 11.3. A Lemma on Near-degenerate Edges 317 11.4. A Stable Local MDS for <S^(A) 318 11.5. A Stable MDS for Splines on a Cell 325 11.6. A Stable Local MDS for S^P(A) 327 11.7. Stability and Local Linear Independence 328 11.8. Remarks 331 11.9. Historical Notes 333 Chapter 12. Bivariate Box Splines 12.1. Type-I Box Splines 334 12.2. Type-II Box Splines 343 12.3. Box Spline Series 347 12.4. The Strang-Fix Conditions 351 12.5. Polynomial Reproducing Formulae 355 12.6. Box Spline Quasi-interpolants 359 12.7. Half Box Splines 363 12.8. Finite Shift-invariant Spaces 366 12.9. Remarks 375 12.10. Historical Notes 377 Chapter 13. Spherical Splines 13.1. Spherical Polynomials 378 13.2. Derivatives of Spherical Polynomials 391 13.3. Spherical Triangulations 396 13.4. Spaces of Spherical Splines 397 13.5. Spherical Macro-element Spaces 406 13.6. Remarks 407 13.7. Historical Notes 408 Chapter 14. Approximation Power of Spherical Splines 14.1. Radial Projection 409 14.2. Projections of Triangulations 409 14.3. Norms on the Sphere 414 14.4. Spherical Sobolev Spaces 416 14.5. Sobolev Seminorms 419 14.6. Clusters of Spherical Triangles 421 14.7. Local Approximation by Spherical Polynomials 423 14.8. The Markov Inequality for Spherical Polynomials 424

Contents ix 14.9. Spaces with Full Approximation Power 425 14.10. Remarks 432 14.11. Historical Notes 433 Chapter 15. Trivariate Polynomials 15.1. The Space V d 434 15.2. Barycentric Coordinates 435 15.3. Bernstein Basis Polynomials 437 15.4. The B-form of a Trivariate Polynomial 438 15.5. Stability of the B-form 440 15.6. The decasteljau Algorithm 441 15.7. Directional Derivatives 442 15.8. B-coefBcients and Derivatives at a Vertex 443 15.9. B-coefficients and Derivatives on Edges 446 15.10. B-coefficients and Derivatives on Faces 449 15.11. B-Coefficients and Hermite Interpolation 451 15.12. The Markov Inequality on Tetrahedra 452 15.13. Integrals and Inner-products 452 15.14. Conditions for Smooth Joins 453 15.15. Approximation Power in the Maximum Norm 454 15.16. Averaged Taylor Polynomials 455 15.17. Approximation Power in the q-norms 456 15.18. Subdivision 457 15.19. Degree Raising 458 15.20. Remarks 458 15.21. Historical Notes 460 Chapter 16. Tetrahedral Partitions 16.1. Properties of a Tetrahedron 461 16.2. General Tetrahedral Partitions 463 16.3. Regular Tetrahedral Partitions ' ; 464 16.4. Euler Relations 465 16.5. Constructing and Storing Tetrahedral Partitions 469 16.6. Clusters of Tetrahedra 470 16.7. Refinements of Tetrahedral Partitions 472 16.8. Delaunay Tetrahedral Partitions 479 16.9. Remarks 479 16.10 Historical Notes 480 Chapter 17. Trivariate Splines 17.1. C Trivariate Spline Spaces 481 17.2. Spaces of Smooth Splines 483 17.3. Minimal Determining Sets 484 17.4. Approximation Power of Trivariate Spline Spaces 486 17.5. Stable Local Bases 489

x Contents 17.6. Nodal Minimal Determining Sets 490 17.7. Hermite Interpolation 492 17.8. Dimension of Trivariate Spline Spaces 494 17.9. Remarks 499 17.10. Historical Notes 500 Chapter 18. Trivariate Macro-element Spaces 18.1. Introduction 502 18.2. A C 1 Polynomial Macro-element. 503 18.3. A C 1 Macro-element on the Alfeld Split 508 18.4. A C 1 Macro-element on the Worsey-Farin Split 513 18.5. A C 1 Macro-element on the Worsey-Piper Split 517 18.6. A C 2 Polynomial Macro-element 520 18.7. A C 2 Macro-element on the Alfeld Split 524 18.8. A C 2 Macro-element on the Worsey-Farin Split 530 18.9. Another C 2 Worsey-Farin Macro-element 537 18.10. AC 2 Macro-element on the Alfeld-16 Split 544 18.11. AC Polynomial Macro-element 548 18.12. Remarks 557 18.13. Historical Notes 558 References 559 Index 587