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From affine to barycentric coordinates in polytopes

Anna B. Romanowska, Jonathan D. H. Smith, Anna Zamojska-Dzienio

TL;DR

The paper addresses sparse barycentric coordinates for points in convex polytopes by developing a decomposition-based framework that partitions a polytope into simplices of the same dimension. It unifies affine and barycentric viewpoints, leveraging volumetric/areal coordinates within each simplex and zeros for vertices outside the containing simplex to produce unique, sparse representations. The approach extends from polygons to polytopes via pointed decompositions into pyramids and simplices, and provides concrete algorithms for 3D and general $n$-dimensional polytopes through shelling-inspired numbering of facets, vertices, and simplices. This yields a practical method for computing barycentric coordinates with minimal support, complementing existing systems like Gibbs and Wachspress and enabling efficient geometric modeling in higher dimensions.

Abstract

Each point of a simplex is expressed as a unique convex combination of the vertices. The coefficients in the combination are the barycentric coordinates of the point. For each point in a general convex polytope, there may be multiple representations, so its barycentric coordinates are not necessarily unique. There are various schemes to fix particular barycentric coordinates: Gibbs, Wachspress, cartographic, etc. In this paper, a method for producing sparse barycentric coordinates in polytopes will be discussed. It uses a purely algebraic treatment of affine spaces and convex sets, with barycentric algebras. The method is based on a certain decomposition of each finite-dimensional convex polytope into a union of simplices of the same dimension.

From affine to barycentric coordinates in polytopes

TL;DR

The paper addresses sparse barycentric coordinates for points in convex polytopes by developing a decomposition-based framework that partitions a polytope into simplices of the same dimension. It unifies affine and barycentric viewpoints, leveraging volumetric/areal coordinates within each simplex and zeros for vertices outside the containing simplex to produce unique, sparse representations. The approach extends from polygons to polytopes via pointed decompositions into pyramids and simplices, and provides concrete algorithms for 3D and general -dimensional polytopes through shelling-inspired numbering of facets, vertices, and simplices. This yields a practical method for computing barycentric coordinates with minimal support, complementing existing systems like Gibbs and Wachspress and enabling efficient geometric modeling in higher dimensions.

Abstract

Each point of a simplex is expressed as a unique convex combination of the vertices. The coefficients in the combination are the barycentric coordinates of the point. For each point in a general convex polytope, there may be multiple representations, so its barycentric coordinates are not necessarily unique. There are various schemes to fix particular barycentric coordinates: Gibbs, Wachspress, cartographic, etc. In this paper, a method for producing sparse barycentric coordinates in polytopes will be discussed. It uses a purely algebraic treatment of affine spaces and convex sets, with barycentric algebras. The method is based on a certain decomposition of each finite-dimensional convex polytope into a union of simplices of the same dimension.
Paper Structure (16 sections, 8 theorems, 26 equations, 1 figure)

This paper contains 16 sections, 8 theorems, 26 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Barycentric algebras
  3. Affine subspaces and convex subsets of real spaces
  4. Affine spaces
  5. Convex sets
  6. The structure of barycentric algebras
  7. Walls
  8. Simplices
  9. Barycentric coordinates in a polytope
  10. Pointed barycentric coordinates
  11. Volumetric coordinates in a simplex
  12. Areal coordinates
  13. Pointed barycentric coordinates in polygons
  14. Pointed barycentric coordinates in polytopes
  15. Pointed barycentric coordinates in $3$-dimensional polytopes
  16. ...and 1 more sections

Key Result

Theorem 2.3

Modes The class $\mathbf{B}$ of barycentric algebras is the variety axiomatized by the identities of idempotence for $p\in I^\circ$, the identities of skew-commutativity for $p\in I^\circ$, and the identities of skew-associativity for $p,q\in I^\circ$, where $p \circ q:= p+q-pq$. The class $\mathbf{C}$ of convex sets is defined, within the variety $\mathbf{B}$, by the quasi-identities of cance

Figures (1)

  • Figure 1: The lopped cube of Example \ref{['X:LoppdCub']}.

Theorems & Definitions (18)

  • Remark 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Proposition 2.5
  • Lemma 2.6
  • Theorem 2.7
  • Remark 2.8
  • Example 2.9
  • Example 2.10
  • ...and 8 more