Bertrand's Paradox addresses a separate issue but illustrates how subtle issues like this can be http://en.wikipedia.org/wiki/Bertrand_paradox_(probability) . The geometric interpretation that the OP offered in which he noted that the map had to be area preserving is a nice way of visualizing why the given implementation doesn't hold but that's just the start. As for the implementation this is also a great example of when something is hard to do in one coordinate system but trivial in another. Polar coordinates are the way to go.