Linear precision for parametric patches

Abstract

Abstract. We give a precise mathematical formulation for the notions of a parametric patch and linear precision, and establish their elementary properties. We relate linear precision to the geometry of a particular linear projection, giving necessary (and quite restrictive) conditions for a patch to possess linear precision. A main focus is on linear precision for Krasauskas' toric patches, which we show is equivalent to a certain rational map on CP d being a birational isomorphism. Lastly, we establish the connection between linear precision for toric surface patches and maximum likelihood degree for discrete exponential families in algebraic statistics, and show how iterative proportional fitting may be used to compute toric patches.

Biharmonic Volumetric Mapping Using Fundamental Solutions

“…A function has the linear precision property if it can reproduce a linear function exactly: given a set of function values of ðv i Þ ¼ rðv i Þ; v i 2 @ for any linear function r, then ðxÞ ¼ rðxÞ, x 2 [35].…”

Linear Precision for Toric Surface Patches

“…We remark that the notion of linear precision used here and in [5] is more restrictive than typically used in geometric modeling. There, linear precision often means that there are control points in ∆ so that the resulting map ∆ → ∆ is the identity.…”

“…In Section 1, we review definitions and results from [5] about linear precision for toric patches, including Proposition 1.4 which asserts that a toric patch has linear precision if and only if a polynomial associated to the patch defines a toric polar Cremona transformation, showing that Corollary 2 follows from Theorem 1. We also show directly that polynomials associated to Bézier triangles, tensor product patches, and trapezoidal patches define toric polar Cremona transformations.…”

“…We review some definitions and results of [5]. See [3,10,13] for more on toric varieties and their relation to geometric modeling.…”

“…, x n ) with positive coefficients, with every such polynomial corresponding to a toric patch. In [5] it was shown that the toric patch given by F has linear precision if and only if the associated toric polar linear system, T (F ) = x 0 ∂F ∂x 0 , x 1 ∂F ∂x 1 , . .…”