, Volume 33, Issue 2, pp 191-214

Linear precision for parametric patches

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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.

Communicated by Helmut Pottmann.
Work of Sottile supported by NSF CAREER grant DMS-0538734, by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation, and by Peter Gritzmann of the Technische Universität München.