Spline Functions on Triangulations

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

2 Contents Preface xi Chapter 1. Bivariate Polynomials 1.1. Introduction Norms of Polynomials on Triangles 1 "1.3. Derivatives of Polynomials Polynomial Approximation in the Maximum Norm Averaged Taylor Polynomials Polynomial Approximation in the q Norm Approximation on Nonconvex ft Interpolation by Bivariate Polynomials Remarks Historical Notes 17 Chapter 2. Bernstein Bezier Methods for Bivariate Polynomials 2.1. Barycentric Coordinates Bernstein Basis Polynomials The B-form Stability of the B-form Representation The decasteljau Algorithm Directional Derivatives Derivatives at a Vertex Cross Derivatives Computing Coefficients by Interpolation Conditions for Smooth Joins of Polynomials Computing Coefficients From Smoothness The Markov Inequality on Triangles -, Integrals and Inner-products of B-polynomials Subdivision Degree Raising Dual Bases for the Bernstein Basis Polynomials A Quasi-interpolant The Bernstein Approximation Operator Remarks Historical Notes 60 Chapter 3. B-Patches 3.1. Control Nets and Control Surfaces The Convex Hull Property Positivity of B-patches Monotonicity of B-patches 70

3 vi Contents 3.5. Convexity of B-patches Control Surfaces and Subdivision Control Surfaces and Degree Raising Rendering a B-patch Parametric Patches Remarks Historical Notes 84 Chapter 4. Triangulations and Quadrangulations 4.1. Properties of Triangles Triangulations Regular Triangulations Euler Relations Storing Triangulations Constructing Triangulations Clusters of Triangles Refinements of Triangulations Optimal Triangulations Maxmin-Angle Triangulations Delaunay Triangulations Constructing Delaunay Triangulations Type-I and Type-II Triangulations Quadrangulations Triangulated Quadrangulations Nested Sequences of Triangulations Remarks Historical Notes. 124 Chapter 5. Bernstein Bezier Methods for Spline Spaces 5.1. The B-form Representation of Splines Storing, Evaluating and Rendering Splines Control Surfaces and the Shape of Spline Surfaces Dimension and a Local Basis for <S (A) Spaces of Smooth Splines Minimal Determining Sets Approximation Power of Spline Spaces Stable Local Bases Nodal Minimal Determining Sets Macro-element Spaces Remarks Historical Notes 149 Chapter 6. C 1 Macro-element Spaces 6.1. A C 1 Polynomial Macro-element Space A C 1 Clough-Tocher Macro-element Space 155

4 Contents vii 6.3. A C 1 Powell-Sabin Macro-element Space A C 1 Powell-Sabin-12 Macro-element Space A C 1 Quadrilateral Macro-element Space Comparison of C 1 Macro-element Spaces Remarks Historical Notes 173 Chapter 7. C 2 Macro-element Spaces 7.1. AC 2 Polynomial Macro-element space A C 2 Clough-Tocher Macro-element Space A C 2 Powell-Sabin Macro-element Space A C 2 Wang Macro-element Space A C 2 Double Clough-Tocher Macro-element A C 2 Quadrilateral Macro-element Space Comparison of C 2 Macro-element Spaces Remarks Historical Notes 198 Chapter 8. C r Macro-element Spaces 8.1. Polynomial Macro-element Spaces Clough-Tocher Macro-element Spaces CT Spaces with Natural Degrees of Freedom Powell-Sabin Macro-element Spaces PS Spaces with Natural Degrees of Freedom Quadrilateral Macro-element Spaces Remarks Historical Notes. 233 Chapter 9. Dimension of Spline Spaces 9.1. Dimension of Spline Spaces on Cells Dimension of Superspline Spaces on Cells Bounds on the Dimension of 5J(A) Dimension of S r d{/\) for d > 3r Dimension of Superspline Spaces Splines on Type-I and Type-II Triangulations Bounds on the Dimension of Superspline Spaces Generic Dimension The Generic Dimension of &} (A) Remarks Historical Notes 274 Chapter 10. Approximation Power of Spline Spaces Approximation Power C Splines and Piecewise Polynomials Approximation Power of <S (A) for d > 3r

5 viii Contents Approximation Power of <SJ(A) for d < 3r Remarks Historical Notes 306 Chapter 11. Stable Local Minimal Determining Sets Introduction Supersplines on Four-cells A Lemma on Near-degenerate Edges A Stable Local MDS for <S^(A) A Stable MDS for Splines on a Cell A Stable Local MDS for S^P(A) Stability and Local Linear Independence Remarks Historical Notes 333 Chapter 12. Bivariate Box Splines Type-I Box Splines Type-II Box Splines Box Spline Series The Strang-Fix Conditions Polynomial Reproducing Formulae Box Spline Quasi-interpolants Half Box Splines Finite Shift-invariant Spaces Remarks Historical Notes 377 Chapter 13. Spherical Splines Spherical Polynomials Derivatives of Spherical Polynomials Spherical Triangulations Spaces of Spherical Splines Spherical Macro-element Spaces Remarks Historical Notes 408 Chapter 14. Approximation Power of Spherical Splines Radial Projection Projections of Triangulations Norms on the Sphere Spherical Sobolev Spaces Sobolev Seminorms Clusters of Spherical Triangles Local Approximation by Spherical Polynomials The Markov Inequality for Spherical Polynomials 424

6 Contents ix Spaces with Full Approximation Power Remarks Historical Notes 433 Chapter 15. Trivariate Polynomials The Space V d Barycentric Coordinates Bernstein Basis Polynomials The B-form of a Trivariate Polynomial Stability of the B-form The decasteljau Algorithm Directional Derivatives B-coefBcients and Derivatives at a Vertex B-coefficients and Derivatives on Edges B-coefficients and Derivatives on Faces B-Coefficients and Hermite Interpolation The Markov Inequality on Tetrahedra Integrals and Inner-products Conditions for Smooth Joins Approximation Power in the Maximum Norm Averaged Taylor Polynomials Approximation Power in the q-norms Subdivision Degree Raising Remarks Historical Notes 460 Chapter 16. Tetrahedral Partitions Properties of a Tetrahedron General Tetrahedral Partitions Regular Tetrahedral Partitions ' ; Euler Relations Constructing and Storing Tetrahedral Partitions Clusters of Tetrahedra Refinements of Tetrahedral Partitions Delaunay Tetrahedral Partitions Remarks Historical Notes 480 Chapter 17. Trivariate Splines C Trivariate Spline Spaces Spaces of Smooth Splines Minimal Determining Sets Approximation Power of Trivariate Spline Spaces Stable Local Bases 489

7 x Contents Nodal Minimal Determining Sets Hermite Interpolation Dimension of Trivariate Spline Spaces Remarks Historical Notes 500 Chapter 18. Trivariate Macro-element Spaces Introduction A C 1 Polynomial Macro-element A C 1 Macro-element on the Alfeld Split A C 1 Macro-element on the Worsey-Farin Split A C 1 Macro-element on the Worsey-Piper Split A C 2 Polynomial Macro-element A C 2 Macro-element on the Alfeld Split A C 2 Macro-element on the Worsey-Farin Split Another C 2 Worsey-Farin Macro-element AC 2 Macro-element on the Alfeld-16 Split AC Polynomial Macro-element Remarks Historical Notes 558 References 559 Index 587

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