If you take the stereographic projection of a uniform random unit quaternion onto 3-space, then to get a uniform distribution in the ball you need to apply a no

If you take the stereographic projection of a uniform random unit quaternion onto 3-space, then to get a uniform distribution in the ball you need to apply a non-linear scaling function to the radius. I’m not sure this really makes much sense as a way to generate random points in the ball (especially if you care about the details of rounding etc.), but maybe fun for someone to implement. I would expect the rejection method to be noticeably faster. * * * Another cute thing you could do is generate 3 normally distributed random numbers (giving a normally distributed random point in a 3-space), and then apply the appropriate non-linear scaling to the radius.