= Poisson polar projections share their total count
{c}
{title2=$\operatorname{Cov}(N_r(A),N_\omega(C))=\sigma(C)\int_A r^{d-1}\,dr$}
For unit volume intensity in the unit ball of $\mathbb R^d$, polar coordinates give intensity $r^{d-1}dr\,\sigma(d\omega)$. The separate radius and direction images are Poisson processes, but each counts the same random total number of original points. Their count <covariance> is the displayed intensity of the intersection of their two preimages. Conditional on the total count, radii and directions are <independent> samples, which does not imply unconditional <independence> of the two point processes.
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