Rigidity of circle polyhedra in the 2-sphere and of hyperideal polyhedra in hyperbolic 3-space

We generalize Cauchy's celebrated theorem on the global rigidity of convex polyhedra in Euclidean 3-space \mathbb{E}^{3} to the context of circle polyhedra in the 2-sphere \mathbb{S}^{2}. We prove that any two convex and proper nonunitary c -polyhedra with Möbius-congruent faces that are consis...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2019-03, Vol.371 (6), p.4215-4249
Main Authors: Bowers, John C., Bowers, Philip L., Pratt, Kevin
Format: Article
Language:English
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Summary:We generalize Cauchy's celebrated theorem on the global rigidity of convex polyhedra in Euclidean 3-space \mathbb{E}^{3} to the context of circle polyhedra in the 2-sphere \mathbb{S}^{2}. We prove that any two convex and proper nonunitary c -polyhedra with Möbius-congruent faces that are consistently oriented are Möbius congruent. Our result implies the global rigidity of convex inversive distance circle packings in the Riemann sphere, as well as that of certain hyperideal hyperbolic polyhedra in \mathbb{H}^{3}.
ISSN:0002-9947
1088-6850
DOI:10.1090/tran/7483