 Poisson line process

Identify an unoriented line not through the origin by $L(p,\theta )=\{z:z\cdot (\cos \theta ,\sin \theta )=p\}$, with $p>0$. A Poisson point process on this parameter space is a random collection of lines. The measure $cdpd\theta$ is invariant under rotations and translations; signed coordinates $t\in \mathbb{R}$, $0\le \theta <\pi$ give the same measure $cdtd\theta$. The number of lines meeting a disk of radius $r$ is Poisson with mean $2\pi cr$.