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Disk Line Picking

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Disk line picking

Using disk point picking,

x=sqrt(r)costheta
(1)
y=sqrt(r)sintheta
(2)

for r in [0,1], theta in [0,2pi), choose two points at random in a unit disk and find the distribution of distances s between the two points. Without loss of generality, take the first point as (r,theta)=(r_1,0) and the second point as (r_2,theta). Then

s^_=(int_0^1int_0^1int_0^pisqrt(r_1+r_2-2sqrt(r_1r_2)costheta)dr_1dr_2dtheta)/(int_0^1int_0^1int_0^pidr_1dr_2dtheta)
(3)
=1/piint_0^1int_0^1int_0^pisqrt(r_1+r_2-2sqrt(r_1r_2)costheta)dr_1dr_2dtheta
(4)
=(128)/(45pi)
(5)
=0.905414787...
(6)

(OEIS A093070; Uspensky 1937, p. 258; Solomon 1978, p. 36).

DiskLinePickingP

This is a special case of ball line picking with n=2, so the full probability function for a disk of radius R is

P_2(s,R)=(4s)/(piR^2)cos^(-1)(s/(2R))-(2s^2)/(piR^3)sqrt(1-(s^2)/(4R^2))
(7)

(Solomon 1978, p. 129; Mathai 1999, p. 204).

The raw moments of the distribution of line lengths are given by

mu_n^'=int_0^2s^nP_2(s,1)ds
(8)
=(2Gamma(n+3))/((n+2)Gamma(2+1/2n)Gamma(3+1/2n)),
(9)

where Gamma(x) is the gamma function and n>-2. The expected value of 1/r is given by n=-1, giving

mu_(-1)=(16)/(3pi)
(10)

(Solomon 1978, p. 36; Pure et al. ). The first few moments are then

mu_0^'=1
(11)
mu_1^'=(128)/(45pi)
(12)
mu_2^'=1
(13)
mu_3^'=(2048)/(525pi)
(14)
mu_4^'=5/3
(15)
mu_5^'=(16384)/(2205pi)
(16)

(OEIS A093526 and A093527 and OEIS A093528 and A093529). The moments mu_(2n-2)^' that are integers occur at n=1, 2, 6, 15, 20, 28, 42, 45, 66, ... (OEIS A014847), which rather amazingly are exactly the values of n such that n|C_n, where C_n is a Catalan number (E. Weisstein, Mar. 30, 2004).

See also

Ball Line Picking, Circle Line Picking, Circular Sector Line Picking, Disk Triangle Picking, Line Line Picking

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References

Sloane, N. J. A. Sequences A014847, A093070, A093526, A093527, A093528, and A093529 in "The On-Line Encyclopedia of Integer Sequences."Mathai, A. M. An Introduction to Geometrical Probability: Distributional Aspects with Applications. Amsterdam, Netherlands: Gordon and Breach, 1999.Pure, R.; Durran, S.; Tong, F.; Pan, J. "Distance Distribution Between Two Random Points in Arbitrary Polygons." To appear in Math. Meth. Appl. Sci.Solomon, H. Geometric Probability. Philadelphia, PA: SIAM, 1978.Uspensky, J. V. Ch. 12, Problem 5 in Introduction to Mathematical Probability. New York: McGraw-Hill, pp. 257-258, 1937.

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Disk Line Picking

Cite this as:

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

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