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Circular Sector Line Picking

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CircularSectorLinePicking

Circular sector line picking considers the distance between two points chosen independently and uniformly with respect to area in a circular sector of radius 1 and central angle theta, where 0<theta<=2pi. The following table gives approximate and exact values of the mean s^_ for several values of theta (Zerr 1891, p. 138). The case theta=2pi corresponds to disk line picking.

thetas^__(approx.)s^__(exact)OEIS
1/2pi0.47387...(32)/(135pi^2)[6ln(2sqrt(2)-2)-94sqrt(2)+48pi+3]A130204
pi0.70605...(64)/(135)(12pi-23)/(pi^2)A130203
2pi0.90541...(128)/(45pi)A093070

For 0<theta<2pi, the mean is

s^_=(8F(theta))/(45theta^2),

where

F(theta)=(c^3-3c-2)ln(1+s)-(c^3-3c+2)lns+32theta+8s^5-4s^4-(28)/3s^3+4s^2-60s,

s=sin(theta/2), and c=costheta (Zerr 1891, p. 138; P. Le Conte, pers. comm., May 10, 2007). The limit of the mean as theta->0^+ is 4/15, and the mean for any radius R>0 is Rs^_.

See also

Circular Sector, Disk Line Picking

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References

Sloane, N. J. A. Sequences A093070, A130203, and A130204 in "The On-Line Encyclopedia of Integer Sequences."Zerr, G. B. M. "Solution to Problem 11031." Mathematical Questions, with Their Solutions, from the Educational Times 55, 137-138, 1891. https://archive.org/details/mathematicalque11unkngoog/page/n145/mode/1up.

Referenced on Wolfram|Alpha

Circular Sector Line Picking

Cite this as:

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

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