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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. n 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

x=sqrt(1-u^2)costheta
(1)
y=sqrt(1-u^2)sintheta
(2)
z=u
(3)

with u in [-1,1] and theta in [0,2pi). 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 u_1=1, or (0,0,1), while the second may be taken as (0,u_2), or (sqrt(1-u_2^2),0,u_2). The average volume is then

V^_=(int_(-1)^1int_(-1)^1int_(-1)^1int_(-1)^1int_0^piint_0^(2pi)|V|du_2du_3du_4dtheta_3dtheta_4)/(int_(-1)^1int_(-1)^1int_(-1)^1int_(-1)^1int_0^piint_0^(2pi)du_2du_3du_4dtheta_3dtheta_4)
(4)
=1/(16pi^2)int_(-1)^1int_(-1)^1int_(-1)^1int_(-1)^1int_0^piint_0^(2pi)|V|du_2du_3du_4dtheta_3dtheta_4,
(5)

where the vertices are located at (x_i,y_i,z_i) where i=1, ..., 4, and the (signed) volume is given by the determinant

 V=1/(3!)|x_1 y_1 z_1 1; x_2 y_2 z_2 1; x_3 y_3 z_3 1; x_4 y_4 z_4 1|.
(6)

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

 V^_=4/(105)pi
(7)

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

mu_2^'=2/(81)
(8)
mu_4^'=4/(2025)
(9)
mu_6^'=(208)/(893025)
(10)
mu_8^'=(4352)/(130203045).
(11)

See also

Ball Tetrahedron Picking, Cube Tetrahedron Picking, Geometric Probability, Point Picking, Sphere Line Picking, Spherical Code, Tetrahedron

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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.

Referenced on Wolfram|Alpha

Sphere Tetrahedron Picking

Cite this as:

Weisstein, Eric W. "Sphere Tetrahedron Picking." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SphereTetrahedronPicking.html

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