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Square Point Picking


SquarePointPickingRandom

Picking two independent sets of points x and y from a unit uniform distribution and placing them at coordinates (x,y) gives points uniformly distributed over the unit square.

SquarePointPickingDistances

The distribution of distances d from a randomly selected point in the unit square to its center is illustrated above.

The expected distance to the square's center is

d^__(center)=int_0^1int_0^1sqrt((x-1/2)^2+(y-1/2)^2)dxdy
(1)
=1/6P
(2)
=1/6(sqrt(2)+sinh^(-1)1)
(3)
=0.3825978582
(4)

(Finch 2003, p. 479; OEIS A103712), where P is the universal parabolic constant. The expected distance to a fixed vertex is given by

d^__(vertex)=int_0^1int_0^1sqrt(x^2+y^2)dxdy
(5)
=1/3[sqrt(2)+sinh^(-1)1],
(6)

which is exactly twice d^__(center).

The expected distances from the closest and farthest vertices are given by

d^__(closest)=1/(24)[2+sqrt(2)sinh^(-1)1]
(7)
d^__(farthest)=1/(24)[18-4sqrt(10)-8sqrt(2)csch^(-1)2+9sqrt(2)sinh^(-1)1-sqrt(2)sinh^(-1)2].
(8)
SquarePointPicking

Pick N points at randomly in a unit square and take the convex hull H. Let <A> be the expected area of H, <s> the expected perimeter, and <P> the expected number of vertices of H. Then

lim_(N->infty)(N(1-<A>))/(lnN)=8/3
(9)
lim_(N->infty)sqrt(N)(4-<s>)=2sqrt(pi)M
(10)
=(4sqrt(2)pi^2)/([Gamma(1/4)]^2)
(11)
=4.2472965...,
(12)
lim_(N->infty)<P>-8/3lnN=8/3(gamma-ln2)
(13)
=-0.309150708...
(14)

(OEIS A096428 and A096429), where M is the multiplicative inverse of Gauss's constant, Gamma(z) is the gamma function, and gamma is the Euler-Mascheroni constant (Rényi and Sulanke 1963, 1964; Finch 2003, pp. 480-481).

In addition,

lim_(N->infty)N<s>=[8/5(3+4sqrt(2))-(32)/5ln(1+sqrt(2))-(8pi^4)/([Gamma(1/4)]^4)]+I_1+I_2+I_3
(15)
=1.37575...,
(16)

where

I_1=-4int_1^infty(sqrt(1+s^2)-s)phi(s-1)ds
(17)
I_2=1/4int_1^inftyint_1^t(sqrt(1+s^2)-s)(sqrt(1+t^2)-t)psi(t/s-1)s^(-3)dsdt
(18)
I_3=1/8int_1^inftyint_1^infty(sqrt(1+s^2)-s)(sqrt(1+t^2)-t)psi(st-1)dsdt
(19)

and

phi(s)=1/(2(s+1)^2)-1/(4s(s+1))+1/(4s)(tan^(-1)(sqrt(s)))/(sqrt(s))
(20)
psi(s)=(15)/(s^3)+1/(s^2)-((15)/(s^3)+6/(s^2)-1/s)(tan^(-1)(sqrt(s)))/(sqrt(s))
(21)

(Groeneboom 1988; Cabo and Groeneboom 1994; Keane 2000; Finch 2003, p. 481).


See also

Box Integral, Cube Point Picking, Square Line Picking, Unit Square Integral

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References

Bailey, D. H.; Borwein, J. M.; and Crandall, R. E. "Box Integrals." Preprint. Apr. 3, 2006.Cabo, A. J. and Groeneboom, P. "Limit Theorems for Functionals of Convex Hulls." Probab. Th. Related Fields 100, 31-55, 1994.Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 480-481, 2003.Groeneboom, P. "Limit Theorems for Complex Hulls." Probab. Th. Related Fields 79, 327-368, 1988.Heuter, I. "Limit Theorems for the Convex Hull of Random Points in Higher Dimensions." Trans. Amer. Math. Soc. 351, 4337-4363, 1999.Keane, J. "Convex Hull Integrals and the 'Ubiquitous Constant.' " Unpublished note, 2000.Rényi, A. and Sulanke, R. "Über die konvexe Hülle von n zufällig gewählten Punkten, I." Z. Wahrscheinlichkeits 2, 75-84, 1963.Rényi, A. and Sulanke, R. "Über die konvexe Hülle von n zufällig gewählten Punkten, II." Z. Wahrscheinlichkeits 3, 138-147, 1964.Sloane, N. J. A. Sequences A096428, A096429, and A103712 in "The On-Line Encyclopedia of Integer Sequences."

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Square Point Picking

Cite this as:

Weisstein, Eric W. "Square Point Picking." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SquarePointPicking.html

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