Rigid ball-polyhedra in Euclidean 3-space

TL;DR

This paper proves a Cauchy-type rigidity theorem for simple, standard ball-polyhedra in Euclidean 3-space, showing they are locally determined by their inner dihedral angles and structure.

Contribution

It introduces a rigidity theorem for ball-polyhedra, extending classical results to a new class of geometric objects in Euclidean space.

Findings

Simple, standard ball-polyhedra are locally rigid with respect to their inner dihedral angles.

The vertex-edge-face structure of these ball-polyhedra forms a lattice.

The rigidity result generalizes classical polyhedral rigidity theorems.

Abstract

A ball-polyhedron is the intersection with non-empty interior of finitely many (closed) unit balls in Euclidean 3-space. One can represent the boundary of a ball-polyhedron as the union of vertices, edges, and faces defined in a rather natural way. A ball-polyhedron is called a simple ball-polyhedron if at every vertex exactly three edges meet. Moreover, a ball-polyhedron is called a standard ball-polyhedron if its vertex-edge-face structure is a lattice (with respect to containment). To each edge of a ball-polyhedron one can assign an inner dihedral angle and say that the given ball-polyhedron is locally rigid with respect to its inner dihedral angles if the vertex-edge-face structure of the ball-polyhedron and its inner dihedral angles determine the ball-polyhedron up to congruence locally. The main result of this paper is a Cauchy-type rigidity theorem for ball-polyhedra stating that any simple and standard ball-polyhedron is locally rigid with respect to its inner dihedral angles.