1. If this (...) leaves you a bit wondering what multivariate splines might be, I am pleased. For I don’t know myself. Carl de Boor

2. Splines over iterated Voronoi diagrams Gerald Farin

3. Overview Voronoi diagrams Sibson’s interpolant quadratic B-splines quadratic iterated splines the general case

4. History B-splines: 1946 - Schoenberg Finite elements: 1950’s - Zienkiewicz... Simplex splines: 1976 – de Boor Recursion: 1972 – de Boor, Mansfield, Cox Bezier triangles: 1980’s – Sabin, Farin Box splines: 1980’s – de Boor, de Vore B-patches: 1982 – Dahmen, Micchelli, Seidel

5. Voronoi diagrams

9. Sibson’s interpolant

10. Sibson basis function

11. Support

12. Properties linear precision 1D: piecewise linear on boundary(CH): piecewise linear C 1 except data sites, C 0 there not idempotent dimension independent

13. Sibson / de Boor de Boor algorithm: pw linear interpolation. Now: pw linear Sibson

14. Quadratic B-spline functions

22. Quadratic surfaces

28. Reminder: Sibson’s...

29. Quadratic surfaces

30. P.Veerapaneni

31. Quadratic surfaces

32. Properties Linear precision 1D: quadratic B-splines dimension independent C 2 (C 1 at u i ) Local support quadratic reproduction

33. Support / Smoothness

39. Basis function

40. “Tangent planes” P. Veerapaneni

41. “Tangent planes”

43. The general case start: set of sites U 0 iterate Voronoi diagrams U 1...U n-1 assign function values Z 0 at U n-1 insert point v 0 generate (locally) refined Voronoi diagram V 0 find Voronoi diagrams V 1...V n-1 compute Z i at V i ; i= n-1,...,1 result: Point Z n at v 0

44. Surface example

45. polynomial precision

46. 1D cubic