 Affine section of a planar set

For $A\subseteq {\mathbb{R}}^2$ and a nonzero direction $\mathbf{b}$, its affine section is the inverse image of $A$ under the line parametrization $t\mapsto \mathbf{a}+t\mathbf{b}$. The resulting set is a subset of the parameter line, so changing a parametrization can change the section by translation or scaling. The four real parameters can be encoded by a single real using real-number pairing by separated digits. Thus no fixed planar set can realize every real subset as a section, by Cantor's theorem.