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Circle Line Picking

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ChordLength
CircleLinePicking

Circle line picking considers the distance between two points chosen independently and uniformly at random on the circumference of a unit circle. Their chord has length s. By rotational and reflection symmetry, the first point can be taken as (1,0) and the second as (costheta,sintheta), where theta has a uniform distribution on [0,pi]. Then

s(theta)=sqrt(2-2costheta)=2|sin(1/2theta)|.
(1)

The average distance is then given by

s^_=(int_0^pis(theta)dtheta)/(int_0^pidtheta)=4/pi.
(2)
CircleLinePickingProb

The probability density function P_s vanishes outside [0,2] and for 0<s<2 is obtained from

P_s=|(dtheta)/(ds)|P_theta=1/pi1/(sqrt(1-(1/2s)^2)).
(3)

For nonnegative integers n, the raw moments are then

mu_n^'=(int_0^pi[2sin(1/2theta)]^ndtheta)/(int_0^pidtheta)
(4)
=int_0^2(s^nds)/(pisqrt(1-(1/2s)^2))
(5)
=(2^nGamma(1/2(1+n)))/(sqrt(pi)Gamma(1+1/2n)),
(6)

where Gamma(z) is the gamma function. The first few are

mu_2^'=2
(7)
mu_3^'=(32)/(3pi)
(8)
mu_4^'=6
(9)
mu_5^'=(512)/(15pi)
(10)
mu_6^'=20
(11)

(OEIS A000984 and OEIS A093581 and A001803), where the numerators of the odd terms are 4 times OEIS A061549.

The central moments are

mu_2=2-(16)/(pi^2)
(12)
mu_3=(8(48-5pi^2))/(3pi^3)
(13)
mu_4=6+(64(pi^2-36))/(3pi^4),
(14)

giving the skewness and kurtosis excess as

gamma_1=(2sqrt(2)(48-5pi^2))/(3(pi^2-8)^(3/2))
(15)
gamma_2=(-9pi^4+320pi^2-2304)/(6(pi^2-8)^2).
(16)

Bertrand's problem asks for the probability that a chord drawn at random on a circle of radius r has length >=r.

See also

Ball Line Picking, Bertrand's Problem, Circle Covering by Arcs, Circle Point Picking, Circle Triangle Picking, Disk Line Picking

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References

Sheng, T. K. "The Distance Between Two Random Points in Plane Regions." Adv. Appl. Prob. 17, 748-773, 1985. https://doi.org/10.2307/1427086.Sloane, N. J. A. Sequences A000984/M1645, A001803/M2986, A061549, and A093581 in "The On-Line Encyclopedia of Integer Sequences."

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Circle Line Picking

Cite this as:

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

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