Advances in Computational Mathematics

, Volume 33, Issue 2, pp 191–214

Linear precision for parametric patches

Article

DOI: 10.1007/s10444-009-9126-7

Cite this article as:
Garcia-Puente, L.D. & Sottile, F. Adv Comput Math (2010) 33: 191. doi:10.1007/s10444-009-9126-7

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 \({\mathbb C}{\mathbb P}^d\) 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.

Keywords

Tensor product Bézier surfacesTriangular Bézier surface patchesBarycentric coordinatesIterative proportional fitting

Mathematics Subject Classifications (2000)

65D1714M25

Copyright information

© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsSam Houston State UniversityHuntsvilleUSA
  2. 2.Department of MathematicsTexas A&M UniversityCollege StationUSA