planar Poisson line process & angles of inclination

Question

Let $L_j$ denote the lines of a planar Poisson line process. By Example 20 of

http://staffhome.ecm.uwa.edu.au/~00025879/reprints/chapter1-abstract.html http://staffhome.ecm.uwa.edu.au/~00025879/reprints/chapter1.ps

inclination angles $\omega_{j}$ of the lines $L_j$ relative to the $x$-axis are independent and identically distributed with density $\sin(\omega)/2$ on $[0,\pi]$. In particular, the vertical lines are weighted more than horizontal lines.

The R package spatstat has planar random process simulation capabilities. I can generate Poisson lines in a window via rpoisline and determine their inclination angles $\omega_{j}$ via angles.psp (with option directed=FALSE). To avoid edge effects, I restrict attention to only those lines fairly close to the window center. Surprisingly, the angles seem to follow not the sine density, but instead the uniform density on $[0,\pi]$.

What am I missing please? I would be grateful for help, e.g., R, Matlab or Mathematica code giving experimental results that match theoretical prediction.

Thank you very much.

Answer 1

The following code shows the paradox mentioned by @adrian-baddeley experimentally. In a circular sampling window we see a nearly uniform distribution of angles:

library(spatstat)
X  <- rpoisline(400, win = disc())
aa <- angles.psp(X)
plot(density(aa))

In a long thin sampling window we see a nearly sinusodial distribution of angles:

X  <- rpoisline(1000, win = owin(c(0,10), c(0,.01)))
aa <- angles.psp(X)
plot(density(aa))
plot(function(x){0.5*sin(x)},from=0,to=pi,add=TRUE,col="red")

Answer 2

This is a classic paradox.

If you consider the random lines which intersect a given, fixed line, then these random lines have angles which are non-uniformly distributed with probability density proportional to the sine of the incidence angle. If you consider the random lines which intersect a given circle then these random lines have uniformly-distributed orientation angles. In each case the bold text describes a selection or sampling operation, and sampling operations introduce bias.