Affine transformations don't change relative density.

You can think of it this way. There's a density function on the shapes in question. Whenever you transform the 2d space, you have to adjust the density at the same time to preserve "volume" (area times density).

Non-linear transforms, such as interpreting a square as polar coordinates to obtain a disk, will expand or shrink area differently in different parts of the space, which means that if you start with a uniform density, you end up with a non-uniform density. But linear/affine transforms affect area the same everywhere in the space, and so if the density is uniform to begin with, it remains uniform.

Thanks, I should have reminded myself that my intuitions are often non-mathematical. I don't even know how I decided that a change in angle would have a non-linear effect on density.

I also had an intuition that the aspect ratio change would squish the distribution. I guess this aspect ratio doesn't matter for the density distribution of dimensionless points.

But, if doing something like splatting a pixel/sprite at each point coordinate, would the sprite shape need to be transformed to match...?