Uniform random (Monte-Carlo) distribution on unit sphere

Question

I need a clarification with algorithm generating random values for my pet ray-tracer.
I emit rays from one point. And I have the problem with distribution of these rays: I need the distribution to be uniform, but it isn't...

The problem I face now is that the distribution being uniform initially is not uniform after my distortions of the space of results.

So for example, I generate r and t angles if the polar coordinate system. The distribution is not uniform and it cannot be uniform: space close to each pole has much more density of results than, say, close to equator. The reason is pretty clear: I convert uniformly distributed points from cylindrical space to the spherical. And I distort results. The same problem is if I normalize points generated randomly in the cube.

My idea now is this: I want to create a tetrahedron, normalize its vertexes, split each face (triangle) with the point in the middle, normalize it and repeat recursively until I have enough points. Then I "distort" these points a little bit. Then I normalize them again. That's it.

I understand that this method is not pure mathematical Monte-Carlo method itself, because I do not use random distribution in any step except for the last one. And I do not like this solution for this complexity.

Can anyone suggest anything more simple yet still

• random
• uniform
• fast
• simple

Thanks!

EDIT:
I need a fast method, not just the correct one. That's why I'm asking about Monte-Carlo. Answers provided are correct, but not fast. The method with tetrahedron is fast, but not very "random" => incorrect.
I really need something more suitable.

Answer 1

Here's an algorithm that allows you to generate points randomly distributed on the unit sphere.