Let A be a convex body in the plane and A1,..., An be translates of A. Such translates give rise to an intersection graph of A, G = (V, E), with vertices V = {1,..., n} and edges E = {uv | Au ∩Av ≠ ∅}. The subgraph G^′ = (V, E^′) satisfying that E^′ ⊂ E is the set of edges uv for which the interiors of Au and Av are disjoint is a unit distance graph of A. If furthermore G^′ = G, i.e., if the interiors of Au and Av are disjoint whenever u ≠ v, then G is a contact 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 such that 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 regular polygons, we show that the contact graphs of A and B are the same if and only if A and B are equivalent. We prove the same statement for unit distance and intersection graphs.