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Suppose that $A$ is a convex body in the plane and that $A_1,\dots,A_n$ aretranslates of $A$. Such translates give rise to an intersection graph of $A$,$G=(V,E)$, with vertices $V=\{1,\dots,n\}$ and edges $E=\{uv\mid A_u\capA_v\neq \emptyset\}$. The subgraph $G'=(V, E')$ satisfying that $E'\subset E$is the set of edges $uv$ for which the interiors of $A_u$ and $A_v$ aredisjoint is a unit distance graph of $A$. If furthermore $G'=G$, i.e., if theinteriors of $A_u$ and $A_v$ are disjoint whenever $u\neq v$, then $G$ is acontact graph of $A$.  In this paper we study which pairs of convex bodies have the same contact,unit distance, or intersection graphs. We say that two convex bodies $A$ and$B$ are equivalent if there exists a linear transformation $B'$ of $B$ suchthat for any slope, the longest line segments with that slope contained in $A$and $B'$, respectively, are equally long. For a broad class of convex bodies,including all strictly convex bodies and linear transformations of regularpolygons, we show that the contact graphs of $A$ and $B$ are the same if andonly if $A$ and $B$ are equivalent. We prove the same statement for unitdistance and intersection graphs.

Classifying Convex Bodies by their Contact and Intersection Graphs