mathematics - Continuous weighted random distribution, biased toward one end

I am currently contributing to a particle system for our game and developing some emitter shapes.

My uniform random distribution along a line or along a rectangular area works fine - no problem.

But now I would like to have something like a 1 dimensional gradient in this distribution. This would mean for example lower values are more common than higher values.

I don't know what would be appropriate mathematical terms for this problem, so my search skills are rather useless with this one. I need something that is computationally simple, as the particle system needs to be efficient.

Answer

Take a look at this picture:

It shows the process of mapping a (random) value to a curve. Suppose you generate a uniformly-distributed random value X, ranging from 0 to 1. By mapping this value to a curve - or, in other words, using f(X) instead of X - you can skew your distribution in whatever way you like.

In this picture, first curve makes higher values more likely; second makes lower values more likely; and the third one makes values cluster in the middle. The exact formula of the curve is not really important, and can be chosen as you like.

For example, first curve looks a bit like square root, and second - like square. Third one is a bit like cube, only translated. If you consider square root to be too slow, first curve also looks like f(X)=1-(1-X)^2 - an inversion of square. Or a hyperbole: f(X)=2X/(1+X).

As a fourth curve shows, you can simply use a precomputed lookup table. Is looks ugly as a curve, but will probably be good enough for a particle system.

This general technique is very simple and powerful. Whatever distribution you need, just imagine a curve mapping, and you'll devise a formula in no time. Or, if your engine has an editor, just make a visual editor for the curve!