Uniform length distribution in cubic line picking

Question

According to MathWorld, the probability distribution of the length of lines between randomly selected pairs of points inside a cube has a very complicated form. In particular it is not uniform. This distribution relies on the fact that the pairs are chosen uniformly and independently. My question is - is there a way to do a reverse engineering and enforce uniform line lengths density? Namely, say I want the probability distribution of the length of lines between randomly selected pairs of points inside a cube to be uniform, from what probability distribution should I sample my pairs? In particular, I want this sampling to still be independent, but not uniform.