  This paper deals with the intersection point process of a stationary and
isotropic Poisson hyperplane process in $\mathbb{R}^d$ of intensity $t>0$,
where only hyperplanes that intersect a centred ball of radius $R>0$ are
considered. Taking $R=t^{-\frac{d}{d+1}}$ it is shown that this point process
converges in distribution, as $t\to\infty$, to a Poisson point process on
$\mathbb{R}^d\setminus\{0\}$ whose intensity measure has power-law density
proportional to $\|x\|^{-(d+1)}$ with respect to the Lebesgue measure. A bound
on the speed of convergence in terms of the Kantorovich-Rubinstein distance is
provided as well. In the background is a general functional Poisson
approximation theorem on abstract Poisson spaces. Implications on the weak
convergence of the convex hull of the intersection point process and the
convergence of its $f$-vector are also discussed, disproving and correcting
thereby a conjecture of Devroye and Toussaint [J.\ Algorithms 14.3 (1993),
381--394] in computational geometry.
