Generate random points inside a square (trivial), mirror one half by the diagonal (preserves distribution), transform coordinates inside the resulting triangle to the given triangle.
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.
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...?
That's what the article suggests.
Clicking through to SO explains it (but assumes you can read numpy). The `s * u` and `t * v` (where `u` and `v` are vectors) are the transformation from right-triangle (half of square) to triangle (half of parallelgram).
I think that sort of splatting is how I intuitively think about this problem, and likely where I went wrong in thinking there would be a distortion of the density function. I'm not confident whether this actually matters though. Would it still approximate the same distribution regardless of what splat shape is used...?
Affine transformations just scale the probability density by a constant, so a uniform distribution is still uniform.