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

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(n4)Θ(n4). 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))p=(x(p),y(p),z(p)) and q=(x(q),y(q),z(q))q=(x(q),y(q),z(q)) on Π  , if dist(s,p)