Advances in Computational Mathematics

, Volume 33, Issue 2, pp 191–214 | Cite as

Linear precision for parametric patches



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.


Tensor product Bézier surfaces Triangular Bézier surface patches Barycentric coordinates Iterative proportional fitting 

Mathematics Subject Classifications (2000)

65D17 14M25 


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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

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