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What if we restrict our attention to closed manifolds? I guess the difficulty with framing this as an algorithmic problem is that it presupposes some way of sp
by teraflop 12y ago
What if we restrict our attention to closed manifolds?
I guess the difficulty with framing this as an algorithmic problem is that it presupposes some way of specifying the input, and I don't know exactly how that would be done. If your input is a polygonal mesh (or higher-dimensional equivalent) then the problem is easy, but what about things like parametric surfaces?
- panic 12y agoYou probably mean bounded manifolds—the plane is closed (it contains all its limit points). The most general condition is that the manifold M itself has a positive, non-infinite measure m(M) (where m is the measure function). Then you can take the probability density of a set S to be m(S)/m(M). From this perspective, the problem with the plane is that it has an infinite surface area, so you can't divide by it to get a well-defined probability density. For parametric surfaces, here's a mathoverflow post answering your exact question: http://mathoverflow.net/questions/9991/how-can-i-sample-uniformly-from-a-surface http://mathoverflow.net/questions/9991/how-can-i-sample-unif... (ultimately linking to this SIGGRAPH paper: http://www.cs.virginia.edu/~jdl/bib/globillum/arvo01_notes.pdf http://www.cs.virginia.edu/~jdl/bib/globillum/arvo01_notes.p...)