Computing $k$-Crossing Visibility through $k$-levels

Let $\mathcal{A}$ be an arrangement of straight lines in the plane (or planes in $\mathbb{R}^3$). The $k$-crossing visibility of a point $p$ on $\mathcal{A}$ is the set of point $q$ in elements of $\mathcal{A}$ such that the segment $pq$ intersects at most $k$ elements of $\mathcal{A}$. In this paper, we obtain algorithms for computing the $k$-crossing visibility. In particular we obtain $O(n\log n + kn)$ and $O(n\log n + k^2n)$ time algorithms, for arrangements of lines in the plane and planes in $\mathbb{R}^3$; which are optimal for $k=\Omega(\log n)$ and $k=\Omega(\sqrt{\log n})$, respectively. We also introduce another algorithm for computing $k$-crossing visibilities on polygons, which reaches the same asymptotical time as the one presented by Bahoo et al. The ideas introduced in this paper can be easily adapted for obtaining $k$-crossing visibilities on other arrangements whose $(\leq k)$-level is known.