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The inverse problem of convex polygon coordinates

A. B. Romanowska, J. D. H. Smith, A. Zamojska-Dzienio

TL;DR

A direct sparse geometric approach based on chordal decompositions of polygons, along with a symmetrized version which creates what is called cartographic coordinates are introduced, which are compared with Gibbs and Wachspress coordinates.

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.

The inverse problem of convex polygon coordinates

TL;DR

A direct sparse geometric approach based on chordal decompositions of polygons, along with a symmetrized version which creates what is called cartographic coordinates are introduced, which are compared with Gibbs and Wachspress coordinates.

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.
Paper Structure (58 sections, 20 theorems, 119 equations, 5 figures)

This paper contains 58 sections, 20 theorems, 119 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Plan of the paper
  4. Notational conventions
  5. Barycentric algebras
  6. Operations on real numbers and binary digits
  7. Binary operations
  8. Barycentric algebras
  9. The basic definition
  10. Basic examples of barycentric algebras
  11. Which barycentric algebras are convex sets?
  12. The entropic property
  13. Subalgebras and homomorphisms
  14. Subalgebras, walls, and sinks
  15. Homomorphisms of barycentric algebras
  16. ...and 43 more sections

Key Result

Lemma 2.6

A barycentric algebra $(A,I^\circ)$ satisfies the entropic property for $s,t,u,v\in A$ and $p,q\in I^\circ$.

Figures (5)

  • Figure 1: Vectors for planar Wachspress coordinates (cf. FloaterWMVC).
  • Figure 2: Planar Wachspress coordinates with curvature (cf. WarSchHirDes).
  • Figure 3: Truncating a polytope to a simple polytope.
  • Figure 4: Curvature and normals to inscribed polygon edges.
  • Figure 5: The norm of the G$-$W discrepancy vector for the quadrilateral $A$ of Example \ref{['X:WxNeqGbs']}, displayed as a contour plot. The dotted line shows the equator (compare §\ref{['SS:EquaEqua']}) where the Gibbs and Wachspress coordinates agree; i.e., where the norm is zero.

Theorems & Definitions (57)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • Remark 2.8
  • Example 2.9
  • ...and 47 more