Affine section of a planar set ($S_{\mathbf{a},\mathbf{b}}(A)=\{t\in \mathbb{R}:\mathbf{a}+t\mathbf{b}\in A\}$)

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.

Use coding of real sequences by ternary digits to give each real sequence a distinct column coordinate $c(s)\in [0,1/2]$. Put every term of that sequence in its column. The vertical affine section of a planar set at $c(s)$ is exactly the sequence range. Every nonempty countable set has such an enumeration, with repetitions allowed, while any column outside the code interval gives the empty set. This realizes every countable real subset; the other sections are unrestricted.