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)
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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)
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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)
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and the expected distance from a fixed vertex is
(4)
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The expected distance from the closest vertex is
(5)
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while the expected distance from the farthest is
(6)
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Picking points independently and uniformly from a triangle with unit area gives a convex hull with expected area of
(7)
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(8)
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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.