Finding the shortest paths on surfaces by fast global approximation precise local refinement

Finding the shortest path between points on a surface is a challenging global optimization problem. It is difficult to devise an algorithm that is computationally efficient, locally accurate and guarantees to converge to the globally shortest path. In this paper a two stage coarse-to-fine approach for finding the shortest paths is suggested. In the first stage the algorithm of Ref. 10 that combines a 3D length estimator with graph search is used to rapidly obtain an approximation to the globally shortest path. In the second stage the approximation is refined to become a shorter geodesic curve, i.e., a locally optimal path. This is achieved by using an algorithm that deforms an arbitrary initial curve ending at two given surface points via geodesic curvature shortening flow. The 3D curve shortening flow is transformed into an equivalent 2D one that is implemented using an efficient numerical algorithm for curve evolution with fixed end points, introduced in Ref. 9.