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Ball Line Picking

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BallLinePicking

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

P_n(s)=n(s^(n-1))/(R^n)I_x(1/2(n+1),1/2),
(1)

where

x=1-(s^2)/(4R^2)
(2)

and

I_x(p,q)=(B(x;p,q))/(B(p,q))
(3)

is a regularized beta function, with B(x;p,q) is an incomplete beta function and B(p,q) is a beta function (Tu and Fischbach 2000).

The first few are

P_1(s)=1/R-s/(2R)
(4)
P_2(s)=(4s)/(piR^2)cos^(-1)(s/(2R))-(2s^2)/(piR^3)sqrt(1-(s^2)/(4R^2))
(5)
P_3(s)=(3s^2)/(R^3)-(9s^3)/(4R^4)+(3s^5)/(16R^6)
(6)
P_4(s)=(8s^3)/(piR^4)cos^(-1)(s/(2R))-(8s^4)/(3piR^5)(1-(s^2)/(4R^2))^(3/2)-(4s^4)/(piR^5)sqrt(1-(s^2)/(4R^2)).
(7)

The mean line segment lengths for R=1 and the first few dimensions n are given by

s^__1=2/3
(8)
s^__2=(128)/(45pi)
(9)
s^__3=(36)/(35)
(10)
s^__4=(16384)/(4725pi)
(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.

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 N-Dimensional Sphere and Its Applications" 17 Apr 2000. http://arxiv.org/abs/math-ph/0004021.

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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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