point on $M_i$. We connect points of $M_i$ by alternating horizontal and vertical segments; the resulting polyline will be called the *staircase* of $M_i$. See Fig. \[fig:shcut-cover\] in Section \[sec:figures\].

We visit corners of ${{\cal C}}$ on a layer $M_i$ in the decreasing order of their $z$-coordinates. Let $c_j$ denote the $j$-th visited corner on $M_i$. We shoot a horizontal ray in $-x$ direction from $\pi(c_j)$ until it hits either a ray of a previously visited corner $c_l$, $l<j$, or the staircase of $M_{i+d}$. We also shoot a vertical ray from $\pi(c_j)$ until it hits either a ray of a previously visited corner $c_{l'}$, $l'<j$, or the staircase of $M_{i+d}$. We will call the polygon bounded by two ray from $\pi(c_j)$, the rays from $c_l$ and $c_{l'}$ and a portion of the staircase $M_{i+d}$ the *region* of $c_j$.

If the region of $c_j$ contains at least $d^2$ corners of $M_j$ for all $j$ such that $i<j \le i+d$, we add all points of $list(C_j)$ to $P_2$ and say that the region of $c_j$ is empty. Otherwise we divide the region of $c_j$ into at most $d$ rectangles, called rectangles of $c_j$ (or rectangles associated to $c_j$). See Fig. \[fig:shcut-cover\] in 