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Rigidity of nonconvex polyhedra with respect to edge lengths and dihedral angles
- Publication Year :
- 2023
-
Abstract
- We prove that every three-dimensional polyhedron is uniquely determined by its dihedral angles and edge lengths, even if nonconvex or self-intersecting, under two plausible sufficient conditions: (i) the polyhedron has only convex faces and (ii) it does not have partially-flat vertices, and under an additional technical requirement that (iii) any triple of vertices is not collinear. The proof is consistently valid for Euclidean, hyperbolic and spherical geometry, which takes a completely different approach from the argument of the Cauchy rigidity theorem. Various counterexamples are provided that arise when these conditions are violated, and self-contained proofs are presented whenever possible. As a corollary, the rigidity of several families of polyhedra is also established. Finally, we propose two conjectures: the first suggests that Condition (iii) can be removed, and the second concerns the rigidity of spherical nonconvex polygons.<br />Comment: 27 pages, 20 figures
- Subjects :
- Mathematics - Geometric Topology
Mathematics - Metric Geometry
52B10, 52C25, 05C10
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2307.14769
- Document Type :
- Working Paper