The Discrete Geodesic Problem

Abstract

We present an algorithm for determining the shortest path between a source and a destination on an arbitrary (possibly nonconvex) polyhedral surface. The path is constrained to lie on the surface, and distances are measured according to the Euclidean metric. Our algorithm runs in time O(n log n) and requires O(n2) space, where n is the number of edges of the surface. After we run our algorithm, the distance from the source to any other destination may be determined using standard techniques in time O(log n) by locating the destination in the subdivision created by the algorithm. The actual shortest path from the source to a destination can be reported in time O(k + log n), where k is the number of faces crossed by the path. The algorithm generalizes to the case of multiple source points to build the Voronoi diagram on the surface, where n is now the maximum of the number of vertices and the number of sources.

Shortest Path Problems on a Polyhedral Surface

“…Although we have previously mentioned the star unfolding only for a convex polyhedral surface P, the concept generalizes to a non-convex polyhedral surface P N because the star unfolding can still be defined by an angularly ordered set of non-crossing shortest path cuts from the source to every vertex [15,35]. In addition, there are still O(M 3 ) edgelets on P N because a shortest path between each pair of vertices can intersect each edge at most once.…”

“…A popular alternative to precomputing all combinatorial shortest paths is to precompute a shortest path map structure SPM(s) that describes all shortest paths from a fixed source s. In the plane with polygonal obstacles, Hershberger and Suri [30] use the continuous Dijkstra paradigm to support all queries from a fixed source after Θ(M log M ) preprocessing. On a (possibly non-convex) polyhedral surface, Mitchell, Mount, and Papadimitriou [35] use the continuous Dijkstra paradigm to construct SPM(s) by propagating a wavefront over a polyhedral surface in O(M 2 log M ) time and O(M 2 ) space. Chen and Han [15] solve the same polyhedral surface problem in O(M 2 ) time and space by combining unfolding and Voronoi diagram techniques.…”

“…This follows because the sum of the eleven angles incident to a is greater than 2π. Thus, the core of the star unfolding for a non-convex polyhedral surface can overlap itself.Even though the star unfolding of a non-convex polyhedral surface can overlap itself, the star unfolding can still be defined by an angularly ordered set of noncrossing shortest path cuts from the source to every vertex [15,35], and a core can still be defined by a polygonal equator with O(M ) complexity that connects adjacent endpoints in the ordering.Since a shortest path can only turn at vertices in the star unfolding [35], an anti-core region must have an hourglass shape [27,28,18]. This follows because an anti-core region is bounded by a point or line segment source, a line segment, and two shortest paths in the unfolded plane.…”

“…Even though the star unfolding of a non-convex polyhedral surface can overlap itself, the star unfolding can still be defined by an angularly ordered set of noncrossing shortest path cuts from the source to every vertex [15,35], and a core can still be defined by a polygonal equator with O(M ) complexity that connects adjacent endpoints in the ordering.…”

“…In the plane with polygonal obstacles, a shortest path can only turn at obstacle vertices, so a shortest path can be combinatorially described as a sequence of obstacle vertices [28]. On a polyhedral surface, a shortest path need not turn at vertices [35], so a path is often described combinatorially by an edge sequence that represents the sequence of edges encountered by the path [1]. A benefit of representing shortest paths by edge sequences is that a series of unfolding rotations can be used to reduce the problem of computing a shortest path on a polyhedral surface into a two-dimensional problem.…”