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

$${\bf a} = \left[\begin{array}{c}\cos\theta_1-1\\ \sin\theta_1\\ 0\end{array}\right]$$ (1)

$${\bf b} = \left[\begin{array}{c}\cos\theta_2\sin\phi_2-1\\ \sin\theta_2\sin\phi_2\\ \cos\phi_2\end{array}\right]$$ (2)

$${\bf c} = \left[\begin{array}{c}\cos\theta_3\sin\phi_3-1\\ \sin\theta_3\sin\phi_3\\ \cos\phi_3\end{array}\right].$$ (3)

The average volume is then

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

where

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

and

${\bf a}\cdot({\bf b}\times{\bf c}) =-\cos\phi_2\sin\theta_1+\cos\phi_3\sin\theta_1$
$-\cos\phi_3\cos\theta_2\sin\phi_2\sin\theta_1+\cos\phi_2\cos\theta_3\sin\phi_3\sin\theta_1$
$-\cos\phi_3\sin\phi_2\sin\theta_2+\cos\phi_3\cos\theta_1\sin\phi_2\sin\theta_2$
$+\cos\phi_2\sin\phi_3\sin\theta_3-\cos\phi_2\cos\theta_1\sin\phi_3\sin\theta_3.\quad$	(6)

The integrals are difficult to compute analytically, but 10⁷ computer Trials give

$$\left\langle{V}\right\rangle{} \approx 0.1080$$ (7)

$$\left\langle{V^2}\right\rangle{} \approx 0.02128$$ (8)

$${\sigma_V}^2 = \left\langle{V^2}\right\rangle{}-\left\langle{V}\right\rangle{}^2 \approx 0.009937.$$ (9)

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

References

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