Shortest paths on polyhedral surfaces
An algorithm is presented that finds the shortest path between two points on a polyhedral surface in O(n5) time, where n is the number of vertices on the surface, thereby establishing that the problem can be solved in polynomial time. The path is confined to the surface, and shortest is defined in terms of Euclidean distance. The algorithm is an extension of methods developed by Sharir and Schorr for finding the shortest path on a convex polyhedron. We relax the convexity assumption, only requiring that the surface be orientable.
KeywordsShort Path Convex Polyhedron Klein Bottle Straight Line Distance Convexity Assumption
Unable to display preview. Download preview PDF.
- [G]P. J. Giblin, Graphs, Surfaces, and Homology, Wiley, New York, 1977.Google Scholar
- [LP]D. T. Lee and F. P. Preparata, "Euclidean shortest paths in the presence of rectilinear boundaries," Proc. 7th Conf. on Graphtheoretical Concepts in Computer Science (1981), 303–316.Google Scholar
- [LW]T. Lozano-Perez and M. A. Wesley, "An algorithm for planning collision-free paths among polyhedral obstacles," CACM, 22 (1979) 560–570.Google Scholar
- [M]D. M. Mount, "On finding shortest paths on convex polyhedra," in preparation (1984).Google Scholar
- [SS]M. Sharir and A. Schorr, "On shortest paths in polyhedral spaces," Proc. 16th Symp. on Theory of Computing (1984), 144–153; full version: Technical Report 84–001, Tel Aviv University, Mar. 1984.Google Scholar