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) [7] proved that the total number of such equivalence classes is Θ ( n 4 ) . 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 p = ( x ( p ) , y ( p ) , z ( p ) ) and q = ( x ( q ) , y ( q ) , z ( q ) ) on Π, if dist ( s , p ) < dist ( s , q ) then z ( p ) ⩾ z ( q ) . Here dist ( s , p ) 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 Θ ( n 4 ) . We also discuss the difficulty of finding the number of equivalence classes on a convex polyhedron.