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Published byBaldwin Ward Modified over 9 years ago
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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
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Splines over iterated Voronoi diagrams Gerald Farin
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Overview Voronoi diagrams Sibson’s interpolant quadratic B-splines quadratic iterated splines the general case
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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
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Voronoi diagrams
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Sibson’s interpolant
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Sibson basis function
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Support
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Properties linear precision 1D: piecewise linear on boundary(CH): piecewise linear C 1 except data sites, C 0 there not idempotent dimension independent
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Sibson / de Boor de Boor algorithm: pw linear interpolation. Now: pw linear Sibson
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Quadratic B-spline functions
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Quadratic surfaces
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Reminder: Sibson’s...
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Quadratic surfaces
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P.Veerapaneni
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Quadratic surfaces
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Properties Linear precision 1D: quadratic B-splines dimension independent C 2 (C 1 at u i ) Local support quadratic reproduction
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Support / Smoothness
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Basis function
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“Tangent planes” P. Veerapaneni
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“Tangent planes”
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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
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Surface example
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polynomial precision
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1D cubic
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