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.