Finding all shortest path edge sequences on a convex polyhedron

Abstract

In this paper, the problems of computing the Euclidean shortest path between two points on the surface of a convex polyhedron and finding all shortest path edge sequences are considered. We propose an O(n6logn) algorithm to find All Shortest Path Edge Sequences, construct n Edge Sequence Trees, and draw out n(n−1)/2 Visibility Relation Diagrams for a given convex polyhedron. According to these data structures, not only can we enumerate all shortest path edge sequences and draw out all maximal ones, but we can also find the shortest path between any two points lying on edges in O(k+logn) time where k is the number of edges crossed by the shortest path.