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Exact Geodesics and Shortest Paths on Polyhedral Surfaces
June 2009 (vol. 31 no. 6)
pp. 1006-1016
Mukund Balasubramanian, Boston University, Boston
Jonathan R. Polimeni, Harvard Medical School, Charlestown
Eric L. Schwartz, Boston University, Boston
We present two algorithms for computing distances along convex and non-convex polyhedral surfaces. The first algorithm computes exact minimal-geodesic distances and the second algorithm combines these distances to compute exact shortest-path distances along the surface. Both algorithms have been extended to compute the exact minimal-geodesic paths and shortest paths. These algorithms have been implemented and validated on surfaces for which the correct solutions are known, in order to verify the accuracy and to measure the run-time performance, which is cubic or less for each algorithm. The exact-distance computations carried out by these algorithms are feasible for large-scale surfaces containing tens of thousands of vertices, and are a necessary component of near-isometric surface flattening methods that accurately transform curved manifolds into flat representations.

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Index Terms:
Curve, surface, solid, and object representations, Differential geometry, Flat maps, Triangular meshes, Surface-based analysis, Computational geometry
Mukund Balasubramanian, Jonathan R. Polimeni, Eric L. Schwartz, "Exact Geodesics and Shortest Paths on Polyhedral Surfaces," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 31, no. 6, pp. 1006-1016, June 2009, doi:10.1109/TPAMI.2008.213
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