  Let $\mathcal{V}$ and $\mathcal{U}$ be the point sets of two independent
homogeneous Poisson processes on $\mathbb{R}^d$. A graph
$\mathcal{G}_\mathcal{V}$ with vertex set $\mathcal{V}$ is constructed by first
connecting pairs of points $(v,u)$ with $v\in\mathcal{V}$ and $u\in\mathcal{U}$
independently with probability $g(v-u)$, where $g$ is a non-increasing radial
function, and then connecting two points $v_1,v_2\in\mathcal{V}$ if and only if
they have a joint neighbor $u\in\mathcal{U}$. This gives rise to a random
intersection graph on $\mathbb{R}^d$. Local properties of the graph, including
the degree distribution, are investigated and quantified in terms of the
intensities of the underlying Poisson processes and the function $g$.
Furthermore, the percolation properties of the graph are characterized and
shown to differ depending on whether $g$ has bounded or unbounded support.
