Triangle Point Picking

Triangle point picking is the uniform selection by area of a random point inside a nondegenerate triangle. Given a triangle with one vertex at the origin and the others at positions and , one might think that a random point inside the triangle would be given by

(1)

where and are independent uniform variates in the interval . This samples the triangle nonuniformly, concentrating points in the corner.

Picking three independent uniform variates in as proportional trilinear coordinates also does not produce uniform point spacing in the triangle. The resulting area density is greatest at the incenter.

To pick points uniformly distributed inside the triangle, instead pick

(2)

where and are independent uniform variates in the interval , which gives points uniformly distributed in the parallelogram spanned by and (left figure). Keep the points with . The remaining points can either be discarded or transformed by into the triangle interior (right figure).

The expected distance of a point picked at random inside an equilateral triangle of unit side length from the center of the triangle is

(3)

and the expected distance from a fixed vertex is

(4)

The expected distance from the closest vertex is

(5)

while the expected distance from the farthest is

(6)

Picking points independently and uniformly from a triangle with unit area gives a convex hull with expected area of

			(7)
			(8)

where is an integer and is a harmonic number (Buchta 1984, 1986). The first few values are 0, 0, 1/12, 1/6, 43/180, 3/10, 197/560, ... (OEIS A093762 and A002548), with . This is a special case of simplex simplex picking.