Let $S_1, S_2 \subseteq \mathbb{R}^2$ be two *finite* disjoint sets of points in the plane with $\texttt{diam}(S_1) \leq 1$ and $\texttt{diam}(S_2) \leq 1$.

Does there always exist a transformation $f: S_1 \cup S_2 \rightarrow \mathbb{R}^2$ such that:

- $\texttt{diam}(f(S_1)) \leq 1$, $\texttt{diam}(f(S_2)) \leq 1$; and
- for any pair of points $s_1 \in S_1$ and $s_2 \in S_2$ we have $d(s_1,s_2) \leq 1 \iff d(f(s_1), f(s_2)) > 1$, where $d(x,y)$ is the Eucledean distance between $x$ and $y$?