Bijection $\psi$ between $\varepsilon \mathbb{Z}^2$ and $\varphi(\varepsilon \mathbb{Z}^2)$ with bounded distances $d(p,\psi(p))$?

Question

Let $\varphi: \mathbb{R}^2\to\mathbb{R}^2$ be a rigid transformation (translation + rotation) and let $\varepsilon \mathbb{Z}^2 = \{ (\varepsilon \cdot k_1, \varepsilon \cdot k_2 ) \in \mathbb{R}^2 | k_1,k_2\in \mathbb{Z} \}$ for a given $\varepsilon > 0$. Is there a way to find a bijection $\psi$ between $\varepsilon \mathbb{Z}^2$ and $\varphi(\varepsilon \mathbb{Z}^2)$ such that for every $p \in \varepsilon \mathbb{Z}^2$, the distance $d(p,\psi(p))$ is bounded above by some value $M(\varepsilon)$?

Thanks and regards!