A uniformly random point inside a circle

Question

You have a random number generator that produces numbers uniformly between 0 and 1. You want a point that is uniformly distributed inside a disc of radius 1, meaning every region of the disc is hit with probability proportional to its area.

The obvious idea is: draw x and y, use x as the distance from the centre and 2 pi y as the angle.

Why does the obvious idea fail?
Fix it using the same two numbers.
Describe a different method that uses rejection.

Accepted Answer

Use radius = the square root of x and angle = 2 pi y. Why "radius = x" fails. The thin ring at distance r from the centre has area proportional to r, so a uniform point in the disc is more likely to be far from the centre than near it: the probability that the radius is at most r must equal the fraction of the disc's area within r, which is r squared. Choosing the radius uniformly makes small radii far too common, and the points cluster near the centre. The fix. We need a random radius R with P(R at most r) = r squared. If x is uniform on 0 to 1, then P(square root of x at most r) = P(x at mo…

A uniformly random point inside a circle

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