Nonoverlap of the star unfolding

Abstract

The star unfolding of a convex polytope with respect to a point x on its surface is obtained by cutting the surface along the shortest paths from x to every vertex, and flattening the surface on the plane. We establish two main properties of the star unfolding: 1. It does not self-overlap: it is a simple polygon. 2. The ridge tree in the unfolding, which is the locus of points with more than one shortest path from x, is precisely the Voronoi diagram of the images of x, restricted to the unfolding. These two properties permit conceptual simplification of several algorithms concerned with shortest paths on polytopes, and sometimes a worst-case complexity improvement as well: • The construction of the ridge tree (in preparation for shortest-path queries, for instance) can be achieved by an especially simple O(n2) algorithm. This is no worst-case complexity improvement, but a considerable simplification nonetheless. • The exact set of all shortest-path "edge sequences" on a polytope can be found by an algorithm considerably simpler than was known previously, with a time improvement of roughly a factor of n over the old bound of O(n7 log n). • The geodesic diameter of a polygon can be found in O(n9 log n) time, an improvement of the previous best O(n10) algorithm. Our results suggest conjectures on "unfoldings" of general convex surfaces. © 1992 Springer-Verlag New York Inc.