As part of a more complex algorithm, I need a fast method to find random points of the n-sphere, $S^n$, starting with a RNG (random number generator). A simple way to do this (in low dimensions at least) is to select a random point of the (n+1)-ball and normalize it. And to get a random point of the (n+1)-ball select a random point of the (n+1)-cube $[-1,1]^{n+1}$ (by selecting (n+1) points of $[0,1)$ with the RNG and scaling using $x \mapsto 2x-1$) and then use "rejection", i.e., just ignore a point if it is not in the (n+1)-ball. This works fine if n is reasonably small, however for large n the volume of the ball is such a tiny fraction of the volume of the cube that rejection is enormously inefficient. So what is a good alternative approach.