On the number of shortest descending paths on the surface of a convex terrain.

Abstract: The shortest paths on the surface of a convex polyhedron can be grouped into equivalence classes according to the sequences of edges, consisting of n-triangular faces, that they cross. Mount (1990) proved that the total number of such equivalence classes is . In this paper, we consider descending paths on the surface of a 3D terrain. A path in a terrain is called a descending path if the z-coordinate of a point p never increases, if we move p along the path from the source to the target. More precisely, a descending path from a point s to another point t is a path Π such that for every pair of points and on Π, if then . Here denotes the distance of p from s along Π. We show that the number of equivalence classes of the shortest descending paths on the surface of a convex terrain is . We also discuss the difficulty of finding the number of equivalence classes on a convex polyhedron. [Copyright &y& Elsevier]