Disk Point Picking

Disk point picking is uniform sampling of points from a disk with respect to area. To generate random points over the unit disk, it is incorrect to choose and independently from uniform distributions on and , respectively, 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). Here and are independent random variables. For each point, use the same sampled value of in both coordinates. Drawing a separate value for each coordinate does not give a uniform distribution over the disk.

The probability density 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 .

Combining disk point picking with an independent height chosen from a uniform distribution on , where , gives cylinder point picking in the interior of a right circular cylinder.