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

You can also use your intuition.

The area of a circle is pi*r^2

For r=1

This give us an area of pi. Let us assume that we have some kind of function fthat would uniformly distrubute N=10 points inside a circle. The ratio here is 10 / pi

Now we double the area and the number of points

For r=2 and N=20

This gives an area of 4pi and the ratio is now 20/4pi or 10/2pi. The ratio will get smaller and smaller the bigger the radius is, because its growth is quadratic and the N scales linearly.

To fix this we can just say

x = r^2
sqrt(x) = r

If you would generate a vector in polar coordinates like this

length = random_0_1();
angle = random_0_2pi();

More points would land around the center.

length = sqrt(random_0_1());
angle = random_0_2pi();

length is not uniformly distributed anymore, but the vector will now be uniformly distributed.