Disk Triangle Picking

Disk triangle picking concerns the triangle determined by three points , , and distributed independently and uniformly in a unit disk (i.e., in the interior of the unit circle). Then the average area of the triangle determined by these points is

(1)

Using disk point picking, this can be written as

(2)

where

(3)

A trigonometric substitution can then be used to remove the trigonometric functions and split the integral into

(4)

where

			(5)
			(6)

However, the easiest way to evaluate the integral is using Crofton's formula and polar coordinates to yield a mean triangle area

(7)

for unit-radius disks (OEIS A189511), or

(8)

for unit-area disks (OEIS A093587; Woolhouse 1867; Solomon 1978; Pfiefer 1989; Zinani 2003). This problem is very closely related to Sylvester's four-point problem, and can be derived as the limit as of the general polygon triangle picking problem.

The probability density function of the triangle area can be expressed in terms of generalized hypergeometric functions (Mathai 1982; Finch 2010, pp. 4-5).

The raw moments of the triangle area in the unit disk are

(9)

where and is the gamma function (Miles 1971; Grote et al. 2019, p. 145). The first few raw moments, starting with , are 1, , 3/32, , 1/32, .... Changing the radius to multiplies the th raw moment by .

The probability that three random points in a disk form an acute triangle is

(10)

(OEIS A093588; Woolhouse 1886). The problem was generalized by Hall (1982) to -dimensional ball triangle picking, and Buchta (1986) gave closed form evaluations for Hall's integrals.