You 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 posi

You 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...)