  Necessary and sufficient conditions are presented for several families of
planar curves to form a set of stable sampling for the Bernstein space
$\mathcal{B}_{\Omega}$ over a convex set $\Omega \subset \mathbb{R}^2$. These
conditions "essentially" describe the mobile sampling property of these
families for the Paley-Wiener spaces $\mathcal{PW}^p_{\Omega},1\leq p<\infty$.
