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

Think about it this way. If you have a rectangle where one axis is radius and one is angle, and you take the points inside this rectangle that are near radius 0. These will all fall very close to the origin (that is close together on the circle.) However, the points near radius R, these will all fall near the edge of the circle (that is, far apart from each other.)

This might give you some idea of why you are getting this behavior.

The factor that's derived on that link tells you how much corresponding area in the rectangle needs to be adjusted to not depend on the radius once it's mapped to the circle.

Edit: So what he writes in the link you share is, "That’s easy enough to do by calculating the inverse of the cumulative distribution, and we get for r:".

The basic premise is here that you can create a variable with a desired distribution from a uniform by mapping the uniform by the inverse function of the cumulative distribution function of the desired probability density function. Why? Just take it for granted for now, but this is a fact.

Here's my somehwat intuitive explanation of the math. The density function f(r) with respect to r has to be proportional to r itself. Understanding this fact is part of any basic calculus books. See sections on polar area elements. Some other posters have mentioned this.

So we'll call it f(r) = C*r;

This turns out to be most of the work. Now, since f(r) should be a probability density, you can easily see that by integrating f(r) over the interval (0,R) you get that C = 2/R^2 (this is an exercise for the reader.)

Thus, f(r) = 2*r/R^2

OK, so that's how you get the formula in the link.

Then, the final part is going from the uniform random variable u in (0,1) you must map by the inverse function of the cumulative distribution function from this desired density f(r). To understand why this is the case you need to find an advanced probability text like Papoulis probably (or derive it yourself.)

Integrating f(r) you get F(r) = r^2/R^2

To find the inverse function of this you set u = r^2/R^2 and then solve for r, which gives you r = R * sqrt(u)

This totally makes sense intuitively too, u = 0 should map to r = 0. Also, u = 1 shoudl map to r = R. Also, it goes by the square root function, which makes sense and matches the link.