Cauchy's theorem and edge lengths of convex polyhedra

Abstract

In this paper we explore, from an algorithmic point of view, the extent to which the facial angles and combinatorial structure of a convex polyhedron determine the polyhedron-in particular the edge lengths and dihedral angles of the polyhedron. Cauchy's rigidity theorem of 1813 states that the dihedral angles are uniquely determined. Finding them is a significant algorithmic problem which we express as a spherical graph drawing problem. Our main result is that the edge lengths, although not uniquely determined, can be found via linear programming. We make use of significant mathematics on convex polyhedra by Stoker, Van Heijenoort, Gale, and Shepherd. © Springer-Verlag Berlin Heidelberg 2007.