Thank you for your careful reading and patient explanations. Finally, I understand your argument. Yes, you are correct, my original reasoning was flawed. I failed to notice that the linear-transformation theorem applies only to regions of positive measure, such as an area in 2-D, a volume in 3-D, etc. I was trying to apply the theorem to a 1-D circle embedded in 2-D space, and, as you correctly noted, that does not work.

Fortunately, the main idea can be salvaged if we apply the theorem correctly. I rewrote the article to discuss uniform densities inside balls and ellipsoids instead of on the boundary of these regions. You might need to refresh your browser cache (or use Private/Incognito mode) to see the new images, which are now inside ellipses or ellipsoids.