Computing approximate shortest paths on convex polytopes
Pankaj K. Agarwal, Sariel Har-Peled, Meetesh Karia · 2000
The algorithms for computing a shortest path on a polyhedral surface are slow, complicated, and numerically unstable.We have developed and implemented a robust and efficient algorithm for computing approximate shortest paths on a convex polyhedral surface.Given a convex polyhedral surface P in R 3 , two points s, t E P, and a parameter e > 0, it computes a path between s and t on P whose length is at most (1 + c) times the length of the shortest path between those points.It first constructs in time O(n/v~ ) a graph of size O(1/s4), computes a shortest path on this graph, and projects the path onto the surface in O(n/c) time, where n is the number of vertices of P. In the post-processing we have added a heuristic that considerably improves the quality of the resulting path.