Generate a random point within a circle (uniformly)

Question

I need to generate a uniformly random point within a circle of radius R.

I realize that by just picking a uniformly random angle in the interval [0 ... 2π),

Answer 1

It really depends on what you mean by 'uniformly random'. This is a subtle point and you can read more about it on the wiki page here: http://en.wikipedia.org/wiki/Bertrand_paradox_%28probability%29, where the same problem, giving different interpretations to 'uniformly random' gives different answers!

Depending on how you choose the points, the distribution could vary, even though they are uniformly random in some sense.

It seems like the blog entry is trying to make it uniformly random in the following sense: If you take a sub-circle of the circle, with the same center, then the probability that the point falls in that region is proportional to the area of the region. That, I believe, is attempting to follow the now standard interpretation of 'uniformly random' for 2D regions with areas defined on them: probability of a point falling in any region (with area well defined) is proportional to the area of that region.