Abstract
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 \reals 3 , two points s, t ∈ P , and a parameter \eps > 0 , it computes a path between s and t on P whose length is at most (1+\eps) times the length of the shortest path between those points. It constructs in time O(n/\sqrt \eps ) a graph of size O(1/\eps 4 ) , computes a shortest path on this graph, and projects the path onto the surface in O(n/\eps) time, where n is the number of vertices of P . In the postprocessing step we have added a heuristic that considerably improves the quality of the resulting path.
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Received July 25, 2000; revised June 6, 2001.
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Agarwal, P., Har-Peled, S. & Karia, M. Computing Approximate Shortest Paths on Convex Polytopes. Algorithmica 33, 227–242 (2002). https://doi.org/10.1007/s00453-001-0111-x
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DOI: https://doi.org/10.1007/s00453-001-0111-x