Disk Point Picking

To generate random points over the unit disk, it is incorrect to use two uniformly distributed variables and and then take

			(1)
			(2)

Because the area element is given by

(3)

this gives a concentration of points in the center (left figure above).

The correct transformation is instead given by

			(4)
			(5)

(right figure above).

The probability function for distance from the center of a point picked at random in a unit disk is

(6)

The raw moments are therefore given by

(7)

giving a mean distance of .