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arXiv · 2312.00828

From affine to barycentric coordinates in polytopes

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

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Anna B. Romanowska, Jonathan D. H. Smith, Anna Zamojska-Dzienio. 2023-11-30. From affine to barycentric coordinates in polytopes. https://doi.org/10.4064/bc129-10

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