Triangle Point Picking


Given a triangle with one vertex at the origin and the others at positions v_1 and v_2, one might think that a random point inside the triangle would be given by


where A_1 and A_2 are uniform variates in the interval [0,1]. However, as can be seen in the plot above, this samples the triangle nonuniformly, concentrating points in the v_1 corner.


Randomly picking each of the trilinear coordinates from a uniform distribution [0,1] 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


where A_1 and A_2 are uniform variates in the interval [0,1], 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


and the expected distance from a fixed vertex is


The expected distance from the closest vertex is

 d^__(closest vertex)=(4+3ln3)/(12sqrt(3)),

while the expected distance from the farthest is

 d^__(farthest vertex) 

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


where H_n 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.

See also

Disk Point Picking, Simplex Simplex Picking, Square Point Picking, Triangle Interior, Triangle Line Picking, Triangle Triangle Picking

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Buchta, C. "Zufallspolygone in konvexen Vielecken." J. reine angew. Math. 347, 212-220, 1984.Buchta, C. "A Note on the Volume of a Random Polytope in a Tetrahedron." Ill. J. Math. 30, 653-659, 1986.Sloane, N. J. A. Sequences A093762 and A093763 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Triangle Point Picking

Cite this as:

Weisstein, Eric W. "Triangle Point Picking." From MathWorld--A Wolfram Web Resource.

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