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.

Near optimal algorithm for the shortest descending path on the surface of a convex terrain

“…The main ingredients of our results are some very interesting characteristics of SDP on the surface convex terrain. A clever blending of these new characteristics of SDP along with the old characteristics of SDP and SGP in [5,8,[10][11][12] will help us to devise an almost near optimal solution.…”

“…(See [5].) Let f and f be two adjacent faces that share an edge e, p be a point on the face f , and q be another point on the edge e such that z(p) z(q).…”

“…[5].) For a pair of points s and t, π d (s, t) consists of a horizontal path segment P h (s, a) from s to some point a on an edge e, followed by a geodesic path segment π geo (a, t).…”

“…Mount [9] demonstrated that SGPs on the surface of a convex polyhedron can be grouped into Θ(n 4 ) equivalence classes. In a very recent paper Ahmed et al [5] showed that SDPs on the surface of a convex terrain can also be grouped into Θ(n 4 ) equivalence classes. They conjectured that SDPs on the surface of a convex polyhedron too will follow the same bound.…”