Linear precision for toric surface patches

TL;DR

This paper classifies homogeneous polynomials that define Cremona transformations and identifies new toric surface patches with linear precision, expanding the known types beyond traditional tensor product patches and Bézier triangles.

Contribution

It provides a classification of polynomials leading to linear precision in toric surface patches, including new trapezoidal shapes not previously recognized.

Findings

Identified a family of trapezoidal toric patches with linear precision.

Classified polynomials defining Cremona transformations in three variables.

Extended the understanding of geometric modeling patches beyond classical shapes.

Abstract

We classify the homogeneous polynomials in three variables whose toric polar linear system defines a Cremona transformation. This classification also includes, as a proper subset, the classification of toric surface patches from geometric modeling which have linear precision. Besides the well-known tensor product patches and B\'ezier triangles, we identify a family of toric patches with trapezoidal shape, each of which has linear precision. B\'ezier triangles and tensor product patches are special cases of trapezoidal patches.