We define an integral affine structure on $\Sigma\backslash D$, using the covering $\mathcal{U}\cup\mathcal{V}$, the nerve of which is the bipartite (two-colored) graph $\Gamma$: $U_{v}\cap V_{w}$ is non-empty if and only if $\langle v,w\rangle=1$. For a point $q\in U_{v}$ we identify the tangent space $T_{q}(\Sigma\backslash D)$ and the lattice $T^{\mathbb{Z}}_{q}$ in it with the following codimension 1 subspace and sublattice of the pair $(\mathbb{R}^{d},\mathbb{Z}^{d})$:

$$T_{q}=\mathbb{R}^{d}_{v}=\{n\in\mathbb{R}^{d}\ :\ \langle v,n\rangle=0\},\qquad T_{q}^{\mathbb{Z}}=\mathbb{Z}^{d}_{v}=\{n\in\mathbb{Z}^{d}\ :\ \langle v,n\rangle=0\}.$$

For a point $q\in V_{w}$ we identify the tangent space $T_{q}(\Sigma\backslash D)$ and the lattice in it with the $(d-1)$-dimensional quotients

$$T_{q}=\mathbb{R}^{d}/w,\qquad T_{q}^{\mathbb{Z}}=\mathbb{Z}^{d}/w.$$

On the overlap $U_{v}\cap V_{w}$, we define the transition map $f_{vw}:\mathbb{R}^{d}_{v}\rightarrow\mathbb{R}^{d}/w$ to be the restriction to the subspace $\mathbb{R}^{d}_{v}$ of the natural projection $\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}/w$.