Let $Γ$ be an arrangement of Jordan curves in the plane, i.e., simple closed curves in the plane. For any curve $γ\in Γ$, we denote the bounded region enclosed by $γ$ as $\tildeγ$. We say that $Γ$ is non-piercing if for any two curves $α, β\in Γ$, $\tildeα \,\setminus\, \tildeβ$ is connected. A non-piercing arrangement of curves generalizes a set of $2$-intersecting curves in which each pair of curves intersect in at most two points. Snoeyink and Hershberger (``Sweeping Arrangements of Curves'', SoCG '89) proved that if we are given an arrangement $Γ$ of $2$-intersecting curves and a {\em sweep} curve $γ\inΓ$, then the arrangement can be \emph{swept} by $γ$ while always maintaining the $2$-intersecting property of the curves in $Γ$. We generalize the result of Snoeyink and Hershberger to the setting of non-piercing arrangements. Given an arrangement $Γ$ of non-piercing curves, a sweep curve $γ\in Γ$, and a point $P$ in $\tildeγ$, we show that we can continuously shrink $γ$ to $P$ so that throughout the process, the arrangement remains non-piercing (except at a finite set of points in time where $γ$ crosses other curves), and $P$ lies in $\tildeγ$. We show that our arguments can be modified if $P$ lies outside $\tildeγ$, and we want to sweep $γ$ \emph{outwards} so that $P$ lies outside $\tildeγ$, and the arrangement remains non-piercing. As a second contribution, we give an alternate proof of the result of Snoeyink and Hershberger, and give several applications of our results to combinatorial and algorithmic questions including to the \emph{multi-hitting set} problem involving points and non-piercing regions.

翻译：设$Γ$为平面上若尔当曲线的排列，即平面上的简单闭曲线。对于任意曲线$γ\in Γ$，我们将其所围的有界区域记为$\tildeγ$。若对于任意两条曲线$α, β\in Γ$，$\tildeα \,\setminus\, \tildeβ$都是连通的，则称$Γ$是非穿刺的。非穿刺曲线排列推广了$2$-相交曲线集（其中每对曲线至多相交于两点）。Snoeyink与Hershberger（《曲线的扫描排列》，SoCG '89）证明：若给定$2$-相交曲线的排列$Γ$及一条{\em 扫描}曲线$γ\inΓ$，则可通过$γ$对排列进行\emph{扫描}，且始终维持$Γ$中曲线的$2$-相交性质。我们将Snoeyink与Hershberger的结果推广到非穿刺排列的情形。给定非穿刺曲线的排列$Γ$、扫描曲线$γ\in Γ$以及点$P\in \tildeγ$，我们证明可以连续地将$γ$收缩至$P$，使得在整个过程中排列保持非穿刺性（除$γ$与其他曲线相交的有限个时刻外），且$P$始终位于$\tildeγ$内。我们进一步证明，若$P$位于$\tildeγ$外部且需将$γ$向\emph{外}扫描以使$P$保持在$\tildeγ$外部，同时维持排列的非穿刺性，我们的论证方法可相应调整。作为第二项贡献，我们给出了Snoeyink与Hershberger结果的替代证明，并将我们的结果应用于若干组合与算法问题，包括涉及点与非穿刺区域的\emph{多重命中集}问题。