Suppose I want to find the shortest path between two points in $\Bbb{R}^3$ with smooth obstacles in the way? I understand things like Dijkstra's algorithm for shortest paths on a graph. But what about in smooth land? In real life I could stretch a string between the points and try to slowly tighten it, possibly trying different ways of weaving it around the obstacles and stuff. But is there an algorithm that finds (or approximates) a global minimum path when the obstacles are smooth? I have heard of A* but I'm not sure if that is what I want. I think this problem is hard in general but what is used in practice?
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How are the smooth obstacles given? $;$ – Mar 01 '15 at 04:07
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@RickyDemer, Suppose they are given analytically. For instance: Generate random spheres in some region of space at the obstacles. You would have a list of, say, radii and centers and would like to know how to traverse the region. – amcalde Mar 03 '15 at 00:07
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"analytically" $: = :$ "as connected components of real affine varieties" $;;;$ ? $;;;;;;;;$ – Mar 03 '15 at 01:20