Linear precision for parametric patches

TL;DR

This paper provides a detailed mathematical framework for parametric patches and linear precision, especially focusing on toric patches, and connects these concepts to algebraic statistics and computational methods.

Contribution

It introduces a precise formulation of linear precision, relates it to geometric projections, and links toric patches to maximum likelihood estimation in algebraic statistics.

Findings

Linear precision relates to a specific linear projection geometry.

Toric patches have linear precision iff a certain rational map is a birational isomorphism.

Iterative proportional fitting can be used to compute toric patches.

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 presision 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.