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 ¿-space E ¿ to the context of circle polyhedra in the ¾-sphere S ¾ . We prove that any two convex and proper non-unitary c-polyhedra with Möbiuscongruent 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 ¿ .

Almost all circle polyhedra are rigid

“…Because of the well-known, intimate connection between the hyperbolic geometry of the unit ball in E ¿ (the Beltrami-Klein model), the inversive geometry of its ¾-sphere ideal boundary S ¾ , and the projective geometry of RP ¿ in which these models reside, (infinitesimal) rigidity results for circle polyhedra and frameworks have implications for the (infinitesimal) rigidity of hyperbolic polyhedra. These have been articulated in works of Thurston, Rivin, Hogdsen-Rivin, Bao-Bonahon, and our previous paper [2]. We do not take the time to translate the results of this paper to the setting of, for example, strictly hyperideal hyperbolic polyhedra in H ¿ (see the final section of [2]) as we are content with this mere mention of the connection.…”

“…These have been articulated in works of Thurston, Rivin, Hogdsen-Rivin, Bao-Bonahon, and our previous paper [2]. We do not take the time to translate the results of this paper to the setting of, for example, strictly hyperideal hyperbolic polyhedra in H ¿ (see the final section of [2]) as we are content with this mere mention of the connection.…”

“…In [2], the authors of the present work began a study of the rigidity theory of circle frameworks, and in particular of circle polyhedra. There we showed that an analogue of the Cauchy Rigidity Theorem remains true when vertices in E ¿ are replaced by circles in the ¾-sphere S ¾ that are placed in the pattern of a Euclidean polyhedron.…”

Rigidity of inversive distance circle packings revisited

“…For the spherical background geometry, Ma and Schlenker [29] had a counterexample showing that there is in general no rigidity and John C. Bowers and Philip L. Bowers [4] obtained a new construction of their counterexample using the inversive geometry of the 2-sphere. John Bowers, Philip Bowers and Kevin Pratt [5] recently proved the global rigidity of convex inversive distance circle packings in the Riemann sphere. Ge and Jiang [12,13] recently studied the deformation of combinatorial curvature and found a way to search for inversive distance circle packing metrics with constant cone angles.…”

Recognition of affine-equivalent polyhedra by their natural developments

“…It has a significant impact on the development of synthetic geometry and is well presented in scientific [4, Chapter III], [12], [13, Section 23.1], [14, Section 6.4], [40], educational [26, Addition K], [43,Chapter 24], and popular science [1, Chapter 14], [15], [20,Theorem 24.1], [33, Chapter III, §14] literature. Of recent articles using or generalizing Theorem 1, we mention article [16], where the local rigidity of zonohedra is proved in R 3 ; articles [17], [18], [44], where the rigidity of a polyhedron is proved in R 3 provided that it is homeomorphic to the sphere, torus or pretzel and all its faces are unit squares; article [9], in which Theorem 1 carries over to the case of circular polytopes in the 2-sphere S 2 ; and article [36], in which a rather unexpected application of Theorem 1 is given to find a sufficient condition for a convex polyhedron P to realize by means of isometries of the ambient space R 3 all "combinatorial symmetries" of P , i. e., all maps of the natural development of P onto itself that preserve edge lengths. A convex polyhedron can be regarded as a metric space if we put by definition that the distance between any two points is equal to the infimum of the lengths of the curves connecting these points and lying entirely on the polyhedron.…”