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# Square Point Picking

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

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

The expected distance to the square's center is

 (1) (2) (3) (4)

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

 (5) (6)

which is exactly twice .

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

 (7) (8)

Pick points at randomly in a unit square and take the convex hull . Let be the expected area of , the expected perimeter, and the expected number of vertices of . Then

 (9) (10) (11) (12) (13) (14)

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

 (15) (16)

where

 (17) (18) (19)

and

 (20) (21)

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

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 " Unpublished note, 2000.Rényi, A. and Sulanke, R. "Über die konvexe Hülle von zufällig gewählten Punkten, I." Z. Wahrscheinlichkeits 2, 75-84, 1963.Rényi, A. and Sulanke, R. "Über die konvexe Hülle von 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."

## Referenced on Wolfram|Alpha

Square Point Picking

## Cite this as:

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