Drawing Points from a Uniform Distribution

Everyone know how to draw points from a uniform distribution, right? Well, what if you want to draw points uniformly over a circle? You can draw from the bounding square and discard points outside the circle. But, this doesn't work for the n-sphere as n grows large (since the expected number of pseudo-draws increases without bound). The question of drawing uniformly from an elipse recently arose on the scipy mailing list and one participant posted an excellent discussion of how to make it work without discarding any draws. One of the key points is that you need (1) a parameterization, and (2) a transformation of that parameterization which yields a constant Jacobian.