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# Ball Line Picking

Given an -ball of radius , find the distribution of the lengths of the lines determined by two points chosen at random within the ball. The probability distribution of lengths is given by

 (1)

where

 (2)

and

 (3)

is a regularized beta function, with is an incomplete beta function and is a beta function (Tu and Fischbach 2000).

The first few are

 (4) (5) (6) (7)

The mean line segment lengths for and the first few dimensions are given by

 (8) (9) (10) (11)

(OEIS A093530 and A093531 and OEIS A093532 and A093533), corresponding to line line picking, disk line picking, (3-D) ball line picking, and so on.

Ball Picking, Ball Point Picking, Ball Tetrahedron Picking, Ball Triangle Picking, Disk Line Picking, Line Line Picking, Sphere Line Picking

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## References

Kendall, M. G. and Moran, P. A. P. Geometrical Probability. New York: Hafner, 1963.Santaló, L. A. Integral Geometry and Geometric Probability. Reading, MA: Addison-Wesley, 1976.Sloane, N. J. A. Sequences A093530, A093531, A093532, and A093533 in "The On-Line Encyclopedia of Integer Sequences."Tu, S.-J. and Fischbach, E. "A New Geometric Probability Technique for an -Dimensional Sphere and Its Applications" 17 Apr 2000. http://arxiv.org/abs/math-ph/0004021.

## Referenced on Wolfram|Alpha

Ball Line Picking

## Cite this as:

Weisstein, Eric W. "Ball Line Picking." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BallLinePicking.html