Cube Tetrahedron Picking
المؤلف:
Seidov, Z. F.
المصدر:
"Letters: Random Triangle." Mathematica J. 7
الجزء والصفحة:
...
8-2-2020
1043
Cube Tetrahedron Picking
Given four points chosen at random inside a unit cube, the average volume of the tetrahedron determined by these points is given by
 |
(1)
|
where the polyhedron vertices are located at 
where 
, ..., 4, and the (signed) volume is given by the determinant
 |
(2)
|
The integral is extremely difficult to compute, but the analytic result for the mean tetrahedron volume is
 |
(3)
|
(OEIS A093524; Zinani 2003). Note that the result quoted in the reply to Seidov (2000) actually refers to the average volume for tetrahedron tetrahedron picking.
REFERENCES:
Do, K.-A. and Solomon, H. "A Simulation Study of Sylvester's Problem in Three Dimensions." J. Appl. Prob. 23, 509-513, 1986.
Seidov, Z. F. "Letters: Random Triangle." Mathematica J. 7, 414, 2000.
Sloane, N. J. A. Sequence A093524 in "The On-Line Encyclopedia of Integer Sequences."
Zinani, A. "The Expected Volume of a Tetrahedron Whose Vertices are Chosen at Random in the Interior of a Cube." Monatshefte Math. 139, 341-348, 2003.
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