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 $mathcal{PW}^p_{Omega},1leq p<infty$.
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