Distinguishing classes of intersection graphs of homothets or similarities of two convex disks

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