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.

翻译：设 $\mathcal{A}$ 为平面中直线（或 $\mathbb{R}^3$ 中平面）的排列。点 $p$ 在 $\mathcal{A}$ 上的 $k$-交叉可见性是指 $\mathcal{A}$ 的元素中点 $q$ 的集合，使得线段 $pq$ 与 $\mathcal{A}$ 中至多 $k$ 个元素相交。本文我们给出了计算 $k$-交叉可见性的算法。特别地，针对平面中直线排列和 $\mathbb{R}^3$ 中平面排列，我们分别获得了 $O(n\log n + kn)$ 和 $O(n\log n + k^2n)$ 时间复杂度的算法；当 $k=\Omega(\log n)$ 和 $k=\Omega(\sqrt{\log n})$ 时，这些算法分别是最优的。我们还引入了另一种多边形上计算 $k$-交叉可见性的算法，该算法达到了与 Bahoo 等人所提出算法相同的渐近时间复杂度。本文提出的思想可轻松推广至已知 $(\leq k)$ 层的其他排列上的 $k$-交叉可见性计算。