Square Line Picking
المؤلف:
Bailey, D. H.; Borwein, J. M.; Kapoor, V.; and Weisstein, E. W.
المصدر:
"Ten Problems in Experimental Mathematics." Amer. Math. Monthly 113
الجزء والصفحة:
...
13-2-2020
1174
Square Line Picking
Square line picking is the selection of pairs of points (corresponding to endpoints of a line segment) randomly placed inside a square. 
random line segments can be picked in a unit square in the Wolfram Language using the function RandomPoint[Rectangle[], ![<span style=]()
{" src="http://mathworld.wolfram.com/images/equations/SquareLinePicking/Inline2.gif" style="height:15px; width:5px" />n, 2![<span style=]()
}" src="http://mathworld.wolfram.com/images/equations/SquareLinePicking/Inline3.gif" style="height:15px; width:5px" />].
Picking two points at random from the interior of a unit square, the average distance between them is the 
case of hypercube line picking, i.e.,
(OEIS A091505).
The exact probability function is given by
![P(l)=<span style=]() {2l(l^2-4l+pi) for 0<=l<=1; 2l[4sqrt(l^2-1)-(l^2+2-pi)-4tan^(-1)(sqrt(l^2-1))] for 1<=l<=sqrt(2) " src="http://mathworld.wolfram.com/images/equations/SquareLinePicking/NumberedEquation1.gif" style="height:64px; width:435px" /> |
(4)
|
(M. Trott, pers. comm., Mar. 11, 2004), and the corresponding distribution function by
![D(l)=<span style=]() {1/2l^4-8/3l^3+pil^2 for 0<=l<=1; -1/2l^4-4l^2tan^(-1)(sqrt(l^2-1))+4/3(2l^2+1)sqrt(l^2-1)+(pi-2)l^2+1/3 for 1<=l<=sqrt(2). " src="http://mathworld.wolfram.com/images/equations/SquareLinePicking/NumberedEquation2.gif" style="height:115px; width:511px" /> |
(5)
|
From this, the mean distance 
can be computed, as can the variance of lengths,
The statistical median is given by the root of the quartic equation
 |
(8)
|
which is approximately 
.
The 
th raw moment is given for 
, 4, 6, ... as 1/3, 17/90, 29/210, 187/1575, 239/207, ... (OEIS A103304 and A103305).
If, instead of picking two points from the interior of a square, two points are chosen at random on different sides of the unit square, the average distance between two points picked in this manner is
(OEIS A091506; Borwein and Bailey 2003, p. 25; Borwein et al. 2004, p. 66).
REFERENCES:
Bailey, D. H.; Borwein, J. M.; Kapoor, V.; and Weisstein, E. W. "Ten Problems in Experimental Mathematics." Amer. Math. Monthly 113, 481-509, 2006b.
Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.
Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.
Sloane, N. J. A. Sequences A091505, A091506, A103304, and A103305 in "The On-Line Encyclopedia of Integer Sequences."
Trott, M. "The Mathematica Guidebooks Additional Material: Average Distance Distribution." http://www.mathematicaguidebooks.org/additions.shtml#S_1_14.
الاكثر قراءة في نظرية الاعداد
اخر الاخبار
اخبار العتبة العباسية المقدسة
