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Wouldn't that subsequent transform distort the sample distribution? As you make the angle at one vertex more obtuse, the density of the point mapping increases
by saltcured 1y ago
Wouldn't that subsequent transform distort the sample distribution? As you make the angle at one vertex more obtuse, the density of the point mapping increases more there relative to the points near the other, more acute vertices.
- hatthew 1y agoI believe an affine transformation would keep density perfectly consistent across the whole area.
- atq2119 1y agoAffine 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.
- saltcured 1y agoThanks, 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...?