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Disk Line Picking

Using disk point picking,

 (1) (2)

for , , choose two points at random in a unit disk and find the distribution of distances between the two points. Without loss of generality, take the first point as and the second point as . Then

 (3) (4) (5) (6)

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

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

 (7)

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

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

 (8) (9)

where is the gamma function and . The expected value of is given by , giving

 (10)

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

 (11) (12) (13) (14) (15) (16)

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

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 Web Resource. https://mathworld.wolfram.com/DiskLinePicking.html