The inverse problem of convex polygon coordinates

TL;DR

This paper compares Gibbs and Wachspress barycentric coordinates for convex polygons, analyzing their similarities, differences, and algebraic properties, to advance understanding of coordinate representations in convex geometry.

Contribution

It provides a detailed comparison of Gibbs and Wachspress coordinates for convex polygons, including conditions of agreement and algebraic interpretations.

Findings

Gibbs and Wachspress coordinates sometimes coincide for convex polygons.

Gibbs coordinates can be expressed as algebraic functions for polygons with rational vertices.

The paper clarifies the relationship and differences between these coordinate systems.

Abstract

Each convex combination of extreme points of a compact convex set represents a certain point of the convex set. Barycentric coordinates provide solutions to the inverse problem of expressing an element of a compact convex set as a convex combination of a finite number of extreme points of the set. Various approaches to this problem have arisen, in various contexts. The most general solution, namely the Gibbs coordinates based on entropy maximization, actually work in the broader setting of barycentric algebras, which constitute semilattice-ordered systems of convex sets. These coordinates involve exponential functions. For convex polytopes, Wachspress coordinates offer solutions which only involve rational functions. The current paper is primarily focused on convex polygons in the plane. After summarizing the Gibbs and Wachspress coordinates, we identify where they agree, and provide comparisons between them when they do not. With an example, we also show how Gibbs coordinates of a polygon with rational vertices may be construed as algebraic functions.