= Affine section of a planar set
{title2=$S_{\mathbf a,\mathbf b}(A)=\{t\in\mathbb R:\mathbf a+t\mathbf b\in A\}$}
= Affine sections of a planar set
{synonym}
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>.
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