For smooth convex disks A, i.e., convex compact subsets of the plane with non-empty interior, we classify the classes G^hom(A) and G^sim(A) of intersection graphs that can be obtained from homothets and similarities of A, respectively. Namely, we prove that G^hom(A) = G^hom(B) if and only if A and B are affine equivalent, and G^sim(A) = G^sim(B) if and only if A and B are similar.