  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.

Computational Geometry