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Date: 21-1-2021
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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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REFERENCES:
Buchta, C. "A Note on the Volume of a Random Polytope in a Tetrahedron." Ill. J. Math. 30, 653-659, 1986.
Finch, S. "Random Triangles VI." http://algo.inria.fr/csolve/rtg6.pdf. Jan. 7, 2011.
Heinrich, L.; Körner, R.; Mehlhorn, N.; and Muche, L. "Numerical and Analytical Computation of Some Second-Order Characteristics of Spatial Poisson-Voronoi Tessellations." Statistics 31, 235-259, 1998.
Miles, R. E. "Isotropic Random Simplices." Adv. Appl. Prob. 3, 353-382, 1971.
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