Triangle Point Picking

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 uniform variates in the interval . However, as can be seen in the plot above, this samples the triangle nonuniformly, concentrating points in the corner.

Randomly picking each of the trilinear coordinates from a uniform distribution also does not produce a uniform point spacing on in the triangle. As illustrated above, the resulting points are concentrated towards the center.

To pick points uniformly distributed inside the triangle, instead pick

(2)

where and are uniform variates in the interval , which gives points uniformly distributed in a quadrilateral (left figure). The points not in the triangle interior can then either be discarded, or transformed into the corresponding point inside the triangle (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 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 A093763). This is a special case of simplex simplex picking.