Spline Functions on Triangulations

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.

Spis treści:
Preface
1. Bivariate polynomials
2. Bernstein-Bézier methods for bivariate polynomials
3. B-patches
4. Triangulations and quadrangulations
5. Bernstein-Bézier methods for spline spaces
6. C1 Macro-element spaces
7. C2 Macro-element spaces
8. Cr Macro-element spaces
9. Dimension of spline splines
10. Approximation power of spline spaces
11. Stable local minimal determining sets
12. Bivariate box splines
13. Spherical splines
14. Approximation power of spherical splines
15. Trivariate polynomials
16. Tetrahedral partitions
17. Trivariate splines
18. Trivariate macro-element spaces
Bibliography
Index.