05760439

j

05760439 Garcia-Puente, Luis David Sottile, Frank Linear precision for parametric patches. Adv. Comput. Math. 33, No. 2, 191-214 (2010). 2010 Springer, Dordrecht EN tensor product B\'ezier surfaces triangular B\'ezier surface patches barycentric coordinates iterative proportional fitting Zbl 0997.65027

doi:10.1007/s10444-009-9126-7

Summary: 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 {\it R. Krasauskas}' toric patches [Adv. Comput. Math. 17, No.~1--2, 89--113 (2002; Zbl 0997.65027)], 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.