Sphere Tetrahedron Picking

Sphere tetrahedron picking is the selection of quadruples of of points corresponding to vertices of a tetrahedron with vertices on the surface of a sphere. random tetrahedra can be picked on a unit sphere in the Wolfram Language using the function RandomPoint[Sphere[], n, 4].

Pick four points on a sphere. What is the probability that the tetrahedron having these points as polyhedron vertices contains the center of the sphere? In the one-dimensional case, the probability that a second point is on the opposite side of 1/2 is 1/2. In the two-dimensional case, pick two points. In order for the third to form a triangle containing the center, it must lie in the quadrant bisected by a line segment passing through the center of the circle and the bisector of the two points. This happens for one quadrant, so the probability is 1/4. Similarly, for a sphere the probability is one octant, or 1/8.

Pick four points at random on the surface of a unit sphere using

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with and . Now find the distribution of possible volumes of the (nonregular) tetrahedra determined by these points. Without loss of generality, the first point may be taken as , or , while the second may be taken as , or . The average volume is then

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where the vertices are located at where , ..., 4, and the (signed) volume is given by the determinant

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The analytic result is difficult to compute, but the exact result for the mean tetrahedron volume is given by

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(Miles 1971, Heinrich et al. 1998, Finch 2011). The raw moments can be computed more easily for even , giving

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