Straight Skeletons and Mitered Offsets of Nonconvex Polytopes

Franz Aurenhammer, Gernot Walzl

Abstract: We give a concise definition of mitered offset surfaces for nonconvex polytopes in \({\mathbbm {R}}^3\), along with a proof of existence and a discussion of basic properties. These results imply the existence of 3D straight skeletons for general nonconvex polytopes. The geometric, topological, and algorithmic features of such skeletons are investigated, including a classification of their constructing events in the generic case. Our results extend to the weighted setting, to a larger class of polytope decompositions, and to general dimensions. For (weighted) straight skeletons of an n-facet polytope in \({\mathbbm {R}}^d\), an upper bound of \(O(n^d)\) on their combinatorial complexity is derived. It relies on a novel layer partition for straight skeletons, and improves the trivial bound by an order of magnitude for \(d \ge 3\).