Thanks, Rick, for this interesting sequel to your previous article.

I'm not sure I understand your reasoning in section "The uniform distribution is preserved under a linear transformation." The statement in the linked reference (about the uniform distribution being preserved) does not apply to the (d-1)-dimensional sphere S because its d-dimensional volume is zero. Indeed, in the first 2-D example the probability densities near (-4, 0) and (4, 0) are greater than those near (0, 3) and (0, -3), resulting in different expected numbers of random points per unit arc length. Also, in my histograms the marginal distributions of the x- and y-coordinates don't look uniform, but rather U-shaped. (As I don't have a SAS/IML license, I used a data step to reproduce the ellipse dataset. Thanks to the common "MTHybrid" random-number generator I obtained the same 200 points.)