= Intersection point process of a Poisson line process
{title2=$\Phi=\{L\cap L':L,L'\in\Pi,\ L\ne L'\}$}
For an isotropic <Poisson line process> of intensity $c\,dp\,d\theta$, its pair intersections form a locally finite <point process>. Coincident intersections and parallel pairs have <probability> zero. Its <intensity measure> is $\pi c^2$ times area. Indeed the <Poisson factorial moment measure> and the <change of variables> from signed distances to intersection location give
$$
\mathbb E\#(\Phi\cap A)=\frac{c^2|A|}{2}\int_0^\pi\int_0^\pi|\sin(\theta-\phi)|\,d\theta\,d\phi=\pi c^2|A|.
$$
The factor $1/2$ counts unordered line pairs. Shared lines create dependence between intersections; this process is not a <Poisson point process>.
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