From affine to barycentric coordinates in polytopes
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866916668658155520 |
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| author | Romanowska, Anna B. Smith, Jonathan D. H. Zamojska-Dzienio, Anna |
| author_facet | Romanowska, Anna B. Smith, Jonathan D. H. Zamojska-Dzienio, Anna |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_00828 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | From affine to barycentric coordinates in polytopes Romanowska, Anna B. Smith, Jonathan D. H. Zamojska-Dzienio, Anna Metric Geometry Combinatorics 08A99, 52A01, 52B99 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. |
| title | From affine to barycentric coordinates in polytopes |
| topic | Metric Geometry Combinatorics 08A99, 52A01, 52B99 |
| url | https://arxiv.org/abs/2312.00828 |