Length distribution of random secants on a unit square

*To*: mathgroup at smc.vnet.net*Subject*: [mg95712] Length distribution of random secants on a unit square*From*: andreas.kohlmajer at gmx.de*Date*: Sun, 25 Jan 2009 06:53:55 -0500 (EST)

I need to work with the length distribution of random secants (of two random points on the perimeter) on a unit square. It's easy to generate some random data and a histogram. I used the following code (Mathematica 7.0): len = Norm[(First[#] - Last[#])] &; corners = {{0, 0}, {1, 0}, {1, 1}, {0, 1}}; dir = {{1, 0}, {0, 1}, {-1, 0}, {0, -1}}; p[t_] := Block[{n, r}, n = Mod[IntegerPart[t], 4]; r = FractionalPart[t]; corners[[n + 1]] + r dir[[n + 1]] ] Histogram[ Table[len[{p[RandomReal[{0, 4}]], p[RandomReal[{0, 4}]]}], {100000}], PlotRange -> All] The histogram shows a small increase close to 1, a big peak at 1 and some kind of exponential decay to Sqrt[2] (= maximum). Does anybody know how to calculate this distribution exactly? What about moving from a unit square to a random rectangle or a random polygon? Thanks!