Straight Skeletons of Three-Dimensional Polyhedra

TL;DR

This paper explores the properties, algorithms, and complexities of straight skeletons in three-dimensional polyhedra, including voxel-based and general nonconvex shapes, providing new algorithms and complexity bounds.

Contribution

It introduces algorithms for constructing 3D straight skeletons for voxel-based and axis-parallel polyhedra, analyzes their complexity, and discusses ambiguity and complexity issues for general polyhedra.

Findings

Voxel-based skeletons can be constructed efficiently with constant time per voxel.

Straight skeletons of axis-parallel polyhedra have O(n^2) features.

General polyhedra have superquadratic worst-case skeleton complexity.

Abstract

This paper studies the straight skeleton of polyhedra in three dimensions. We first address voxel-based polyhedra (polycubes), formed as the union of a collection of cubical (axis-aligned) voxels. We analyze the ways in which the skeleton may intersect each voxel of the polyhedron, and show that the skeleton may be constructed by a simple voxel-sweeping algorithm taking constant time per voxel. In addition, we describe a more complex algorithm for straight skeletons of voxel-based polyhedra, which takes time proportional to the area of the surfaces of the straight skeleton rather than the volume of the polyhedron. We also consider more general polyhedra with axis-parallel edges and faces, and show that any n-vertex polyhedron of this type has a straight skeleton with O(n^2) features. We provide algorithms for constructing the straight skeleton, with running time O(min(n^2 log n, k log^{O(1)} n)) where k is the output complexity. Next, we discuss the straight skeleton of a general nonconvex polyhedron. We show that it has an ambiguity issue, and suggest a consistent method to resolve it. We prove that the straight skeleton of a general polyhedron has a superquadratic complexity in the worst case. Finally, we report on an implementation of a simple algorithm for the general case.