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

John C. Bowers, Philip L. Bowers, Kevin Pratt

We generalize Cauchy's celebrated theorem on the global rigidity of
convex polyhedra in Euclidean 3-space **E**^{3} to the
context of circle polyhedra in the 2-sphere **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 **H**^{3}.