From affine to barycentric coordinates in polytopes

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 necessaril...

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Bibliographic Details
Published inarXiv.org
Main Authors Romanowska, Anna B, Smith, Jonathan D H, Zamojska-Dzienio, Anna
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 30.11.2023
Subjects
Apexes
Convexity
Mathematical analysis
Polytopes
Production methods
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Summary: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.
ISSN:2331-8422