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

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

(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.

See also 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

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