Square Point Picking
المؤلف:
Bailey, D. H.; Borwein, J. M.; and Crandall, R. E
المصدر:
"Box Integrals." Preprint. Apr. 3, 2006.
الجزء والصفحة:
...
28-8-2018
2496
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
(Finch 2003, p. 479; OEIS A103712), where 
is the universal parabolic constant. The expected distance to a fixed vertex is given by
which is exactly twice 
.
The expected distances from the closest and farthest vertices are given by
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
(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).
In addition,
where
and
(Groeneboom 1988; Cabo and Groeneboom 1994; Keane 2000; Finch 2003, p. 481).
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 " Unpublished note, 2000.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."
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