Efficient Exact Enumeration of Single-Source Geodesics on a Non-Convex Polyhedron

Abstract: In this paper, we consider enumeration of geodesics on a polyhedron, where a geodesic means locally-shortest path between two points. Particularly, we consider the following preprocessing problem: given a point $s$ on a polyhedral surface and a positive real number $r$, to build a data structure that enables, for any point $t$ on the surface, to enumerate all geodesics from $s$ to $t$ whose length is less than $r$. First, we present a naive algorithm by removing the trimming process from the MMP algorithm (1987). Next, we present an improved algorithm which is practically more efficient on a non-convex polyhedron, in terms of preprocessing time and memory consumption. Moreover, we introduce a single-pair geodesic graph to succinctly encode a result of geodesic query. Lastly, we compare these naive and improved algorithms by some computer experiments.