Rigidity of Circle Polyhedra in the 2-Sphere and of Hyperideal Polyhedra in Hyperbolic 3-Space

Abstract

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 non-unitary 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 ${\mathbb{H}}^3$.

Cited by in corpus (2)

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