Short-Cuts on Star, Source and Planar Unfoldings

Abstract

When studying a 3D convex polyhedron, it is often easier to cut it open and flatten in on the plane. There are several ways to perform this unfolding. Standard unfoldings which have been used in literature include Star Unfoldings, Source Unfoldings, and Planar Unfoldings, each differing only in the cuts that are made. Note that every unfolding has the property that a straight line between two points on this unfolding need not be contained completely within the body of this unfolding. This could potentially lead to situations where the above straight line is shorter than the shortest path between the corresponding end points on the convex polyhedron. We call such straight lines short-cuts. The presence of short-cuts is an obstacle to the use of unfoldings for designing algorithms which compute shortest paths on polyhedra. We study various properties of Star, Source and Planar Unfoldings which could play a role in circumventing this obstacle and facilitating the use of these unfoldings for shortest path algorithms.

We begin by showing that Star and Source Unfoldings do not have short-cuts. We also describe a new structure called the Extended Source Unfolding which exhibits a similar property. In contrast, it is known that Planar unfoldings can indeed have short-cuts. Using our results on Star, Source and Extended Source Unfoldings above and using an additional structure called the Compacted Source Unfolding, we provide a necessary condition for a pair of points on a Planar Unfolding to form a short-cut. We believe that this condition could be useful in enumerating all Shortest Path Edge Sequences on a convex polyhedron in an output-sensitive way, using the Planar Unfolding.