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

Splines over iterated Voronoi diagrams Gerald Farin

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

History B-splines: 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

Voronoi diagrams

Sibson’s interpolant

Sibson basis function

Support

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

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

Quadratic B-spline functions

Quadratic surfaces

Reminder: Sibson’s...

Quadratic surfaces

P.Veerapaneni

Quadratic surfaces

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

Support / Smoothness

Basis function

“Tangent planes” P. Veerapaneni

“Tangent planes”

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

Surface example

polynomial precision

1D cubic