= \mathbb{Z}^2$ not crossing these spaces except at the origin. As window we consider the projection into the internal space of the unit square in $\mathbb{R}^2$.\
The point pattern $\mathfrak{P}(W)$ may be written using the $*$-map in the following form $$\begin{aligned}
 \mathfrak{P}(W) := \left\lbrace \gamma \in \Gamma \; \vert \; \gamma ^*\in W\right\rbrace \end{aligned}$$

We may allows ourselves to translate the resulting point pattern by any vector $t\in \mathbb{R}^d$ site by site in the physical space, which we call here *physical translation*, or translate the window $W$ by an element $w\in \mathsf{H}$, which we call *internal translation*. In both cases this leads to a new point pattern of $\mathbb{R}^d$. We now introduce the class of *model sets* of $\mathbb{R}^d$ as follows:

\[def:inter.model.set\] An inter model set $\Lambda$ associated with a cut $\&$ project scheme $(\mathsf{H}, \Sigma , \mathbb{R}^d)$ together with a window $W$ is a subset of $\mathbb{R}^d$ of the form $$\begin{aligned}
 \mathfrak{P}(w+ \stackrel{\circ }{W}) -t \subseteq \Lambda \subseteq \mathfrak{P}(w+ W) -t\end{aligned}$$

An inter model set is called *regular* whenever the window $W$ used to construct it has boundary of Haar measure zero in $\mathsf{H}$. Due to the assumptions on the underlying cut $\&$ project scheme and on the window 