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Tetrahedron Inscribing

Pick four points at random on the surface of a unit Sphere. Find the distribution of possible volumes of (nonregular) Tetrahedra. Without loss of generality, the first point can be chosen as (1, 0, 0). Designate the other points ${\bf a}$, ${\bf b}$, and ${\bf c}$. Then the distances from the first Vertex are

$\displaystyle {\bf a}$ $\textstyle =$ $\displaystyle \left[\begin{array}{c}\cos\theta_1-1\\  \sin\theta_1\\  0\end{array}\right]$ (1)
$\displaystyle {\bf b}$ $\textstyle =$ $\displaystyle \left[\begin{array}{c}\cos\theta_2\sin\phi_2-1\\  \sin\theta_2\sin\phi_2\\  \cos\phi_2\end{array}\right]$ (2)
$\displaystyle {\bf c}$ $\textstyle =$ $\displaystyle \left[\begin{array}{c}\cos\theta_3\sin\phi_3-1\\  \sin\theta_3\sin\phi_3\\  \cos\phi_3\end{array}\right].$ (3)

The average volume is then

\bar V = {1\over C} \int_{0}^{2\pi}\int_0^{2\pi}\int_0^{2\pi...
...f c})\vert\,d\phi_3\,d\phi_2\,d\theta_3\,d\theta_2\,d\theta_1,
\end{displaymath} (4)

C={\int_{0}^{2\pi}\int_0^{2\pi}\int_0^{2\pi}\int_{-\pi/2}^{\pi/2} d\phi_3\,d\phi_2\,d\theta_3\,d\theta_2\,d\theta_1} = 8\pi^5
\end{displaymath} (5)

${\bf a}\cdot({\bf b}\times{\bf c}) =-\cos\phi_2\sin\theta_1+\cos\phi_3\sin\theta_1$
$ -\cos\phi_3\cos\theta_2\sin\phi_2\sin\theta_1+\cos\phi_2\cos\theta_3\sin\phi_3\sin\theta_1$
$ +\cos\phi_2\sin\phi_3\sin\theta_3-\cos\phi_2\cos\theta_1\sin\phi_3\sin\theta_3.\quad$ (6)
The integrals are difficult to compute analytically, but 107 computer Trials give
$\displaystyle \left\langle{V}\right\rangle{}$ $\textstyle \approx$ $\displaystyle 0.1080$ (7)
$\displaystyle \left\langle{V^2}\right\rangle{}$ $\textstyle \approx$ $\displaystyle 0.02128$ (8)
$\displaystyle {\sigma_V}^2$ $\textstyle =$ $\displaystyle \left\langle{V^2}\right\rangle{}-\left\langle{V}\right\rangle{}^2 \approx 0.009937.$ (9)

See also Point-Point Distance--1-D, Triangle Inscribing in a Circle, Triangle Inscribing in an Ellipse


Buchta, C. ``A Note on the Volume of a Random Polytope in a Tetrahedron.'' Ill. J. Math. 30, 653-659, 1986.

© 1996-9 Eric W. Weisstein