Straight Skeletons and Mitered Offsets of Nonconvex Polytopes
Franz Aurenhammer, Gernot Walzl · Discrete & Computational Geometry · 2016
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 $$d \ge 3$$ .