Rigidity of Nonconvex Polyhedra with Respect to Edge Lengths and Dihedral Angles.
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| Title: | Rigidity of Nonconvex Polyhedra with Respect to Edge Lengths and Dihedral Angles. |
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| Authors: | Cho, Yunhi1,2 (AUTHOR) yhcho@uos.ac.kr, Kim, Seonhwa1 (AUTHOR) seonhwa17kim@uos.ac.kr |
| Source: | Discrete & Computational Geometry. Sep2025, Vol. 74 Issue 2, p302-336. 35p. |
| Subjects: | Dihedral angles, Polyhedra, Hypothesis, Geometric rigidity, Geometry |
| 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. [ABSTRACT FROM AUTHOR] |
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| Database: | Engineering Source |
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| 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. [ABSTRACT FROM AUTHOR] |
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| ISSN: | 01795376 |
| DOI: | 10.1007/s00454-024-00670-w |
