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19 June 2009 ArXiv org 0806 3230 LINEAR PRECISION FOR TORIC SURFACE PATCHES HANS CHRISTIAN GRAF VON BOTHMER KRISTIAN RANESTAD AND FRANK SOTTILE Abstract We classify the homogeneous polynomials in three variables whose toric polar linear system defines a Cremona transformation This classification includes as a proper subset the classification of toric surface patches from geometric modeling which have linear precision Besides the well known tensor product patches and Be zier triangles we identify a family of toric patches with trapezoidal shape each of which has linear precision Furthermore Be zier triangles and tensor product patches are special cases of trapezoidal patches Communicated by Wolfgang Dahmen and Herbert Edelsbrunner Introduction While the basic units in the geometric modeling of surfaces are Be zier triangles and rectangular tensor product patches some applications call for multi sided C patches see 8 for a discussion Krasauskas s toric Be zier patches 10 are a flexible and mathematically appealing system of such patches These are based on real toric varieties from algebraic geometry may have shape any polytope with integer vertices and they include the classical Be zier patches as special cases For descriptions of multisided patches and toric patches see 6 More precisely given a set of lattice points in Zn with convex hull Krasauskas defined toric Be zier functions which are polynomial blending functions associated to each lattice point This collection of lattice points and toric Be zier functions together with a positive weight associated to each lattice point is a toric patch Choosing also a control point in Rd for each lattice point leads to a map Rd whose image may be used in modeling If we choose the lattice points themselves as control points we obtain the tautological map which is a bijection If the tautological map has a rational inverse then the toric patch has linear precision The lattice points and weights of a toric patch are encoded in a



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