Rigidity, Tensegrity and Reconstruction of Polytopes under Metric Constraints
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866910288876404736 |
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| author | Winter, Martin |
| author_facet | Winter, Martin |
| contents | We conjecture that a convex polytope is uniquely determined up to isometry by its edge-graph, edge lengths and the collection of distances of its vertices to some arbitrary interior point, across all dimensions and all combinatorial types. We conjecture even stronger that for two polytopes $P\subset\mathbb R^d$ and $Q\subset\mathbb R^e$ with the same edge-graph it is not possible that $Q$ has longer edges than $P$ while also having smaller vertex-point distances.
We develop techniques to attack this question and verify it in three relevant special cases: if $P$ and $Q$ are centrally symmetric, if $Q$ is a slight perturbation of $P$, and if $P$ and $Q$ are combinatorially equivalent. In the first two cases the statements stay true if we replace $Q$ by some graph embedding $q\colon V(G_P)\to\mathbb R^e$ of the edge-graph $G_P$ of $P$, which can be interpreted as local resp. universal rigidity of certain tensegrity frameworks. We also establish that a polytope is uniquely determined up to affine equivalence by its edge-graph, edge lengths and the Wachspress coordinates of an arbitrary interior point.
We close with a broad overview of related and subsequent questions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_14194 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rigidity, Tensegrity and Reconstruction of Polytopes under Metric Constraints Winter, Martin Combinatorics Metric Geometry 51M20, 52B11, 52C25, 05C50 We conjecture that a convex polytope is uniquely determined up to isometry by its edge-graph, edge lengths and the collection of distances of its vertices to some arbitrary interior point, across all dimensions and all combinatorial types. We conjecture even stronger that for two polytopes $P\subset\mathbb R^d$ and $Q\subset\mathbb R^e$ with the same edge-graph it is not possible that $Q$ has longer edges than $P$ while also having smaller vertex-point distances. We develop techniques to attack this question and verify it in three relevant special cases: if $P$ and $Q$ are centrally symmetric, if $Q$ is a slight perturbation of $P$, and if $P$ and $Q$ are combinatorially equivalent. In the first two cases the statements stay true if we replace $Q$ by some graph embedding $q\colon V(G_P)\to\mathbb R^e$ of the edge-graph $G_P$ of $P$, which can be interpreted as local resp. universal rigidity of certain tensegrity frameworks. We also establish that a polytope is uniquely determined up to affine equivalence by its edge-graph, edge lengths and the Wachspress coordinates of an arbitrary interior point. We close with a broad overview of related and subsequent questions. |
| title | Rigidity, Tensegrity and Reconstruction of Polytopes under Metric Constraints |
| topic | Combinatorics Metric Geometry 51M20, 52B11, 52C25, 05C50 |
| url | https://arxiv.org/abs/2302.14194 |