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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: 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
Voronoi diagrams
Voronoi diagrams
Voronoi diagrams
Voronoi diagrams
Sibson’s interpolant
Sibson basis function
Support
Properties
• linear precision
• 1D: piecewise linear
• on boundary(CH): piecewise linear
• C1 except data sites, C0 there
• not idempotent
• dimension independent
Sibson / de Boor
de Boor algorithm: pw linear interpolation.
Now:
pw linear Sibson
Quadratic B-spline functions
Quadratic B-spline functions
Quadratic B-spline functions
Quadratic B-spline functions
Quadratic B-spline functions
Quadratic B-spline functions
Quadratic B-spline functions
Quadratic B-spline functions
Quadratic surfaces
Quadratic surfaces
Quadratic surfaces
Quadratic surfaces
Quadratic surfaces
Quadratic surfaces
Reminder: Sibson’s...
Quadratic surfaces
Quadratic surfaces
P.Veerapaneni
Quadratic surfaces
Properties
• Linear precision
• 1D: quadratic B-splines
• dimension independent
• C2 (C1 at ui)
• Local support
• quadratic reproduction
Support / Smoothness
Support / Smoothness
Support / Smoothness
Support / Smoothness
Support / Smoothness
Support / Smoothness
Basis function
“Tangent planes”
P. Veerapaneni
“Tangent planes”
“Tangent planes”
The general case
• start: set of sites U0
• iterate Voronoi diagrams U1...Un-1
• assign function values Z0 at Un-1
• insert point v0
• generate (locally) refined Voronoi diagram V0
• find Voronoi diagrams V1...Vn-1
• compute Zi at Vi; i= n-1,...,1• result: Point Zn at v0
Surface example
polynomial precision
1D cubic
1D cubic
1D cubic
1D cubic
1D cubic
1D cubic
1D cubic
1D cubic