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 x = a_1 v_1 + (1 - a_1) a_2 v_2, 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.
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