Toric degenerations of Bezier patches

TL;DR

This paper introduces regular control surfaces as a new, non-polyhedral control structure for Bezier surface patches, describing their role as limits under varying weights and their geometric significance.

Contribution

It proposes regular control surfaces as a novel, canonical limiting structure for Bezier patches, extending the concept of control polygons from curves to surfaces.

Findings

Regular control surfaces are the limits of Bezier patches with varying weights.

They provide a geometric interpretation similar to control polygons for curves.

The concept generalizes the control structure for surface patches.

Abstract

The control polygon of a Bezier curve is well-defined and has geometric significance---there is a sequence of weights under which the limiting position of the curve is the control polygon. For a Bezier surface patch, there are many possible polyhedral control structures, and none are canonical. We propose a not necessarily polyhedral control structure for surface patches, regular control surfaces, which are certain C^0 spline surfaces. While not unique, regular control surfaces are exactly the possible limiting positions of a Bezier patch when the weights are allowed to vary.