Weakly Inscribed Polyhedra
| Autor: | Hao Chen, Jean-Marc Schlenker |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
Mathematics - Differential Geometry
Mathematics - Geometric Topology Mathematics - Metric Geometry Differential Geometry (math.DG) FOS: Mathematics Mathematics [G03] [Physical chemical mathematical & earth Sciences] Mathematics::Metric Geometry Metric Geometry (math.MG) Geometric Topology (math.GT) Mathématiques [G03] [Physique chimie mathématiques & sciences de la terre] General Medicine Computer Science::Computational Geometry |
| DOI: | 10.48550/arxiv.1709.10389 |
| Popis: | We study convex polyhedra in $\mathbb{R}\mathbb{P}^3$ with all their vertices on a sphere. We do not require, in particular, that the polyhedra lie in the interior of the sphere, hence the term "weakly inscribed". Such polyhedra can be interpreted as ideal polyhedra, if we regard $\mathbb{R}\mathbb{P}^3$ as a combination of the hyperbolic space and the de Sitter space, with the sphere as the common ideal boundary. We have three main results: (1) the $1$-skeleta of weakly inscribed polyhedra are characterized in a purely combinatorial way, (2) the exterior dihedral angles are characterized by linear programming, and (3) we also describe the hyperbolic-de Sitter structure induced on the boundary of weakly inscribed polyhedra. Comment: 25 pages, 12 figures. v2: improved exposition, etc. v3: mostly updated introduction for clarity |
| Databáze: | OpenAIRE |
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