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 , , and . Then the distances from the first Vertex are

			(1)
			(2)
			(3)

(4)

(5)

	(6)

⁷ computer Trials give

			(7)
			(8)
			(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.

• Reptend Prime
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