Let $\Gamma$ be a finite set of Jordan curves in the plane. For any curve $\gamma \in \Gamma$, we denote the bounded region enclosed by $\gamma$ as $\tilde{\gamma}$. We say that $\Gamma$ is a non-piercing family if for any two curves $\alpha , \beta \in \Gamma$, $\tilde{\alpha} \setminus \tilde{\beta}$ is a connected region. A non-piercing family of curves generalizes a family 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 a family $\mathcal{C}$ of $2$-intersecting curves and a fixed curve $C\in\mathcal{C}$, then the arrangement can be \emph{swept} by $C$, i.e., $C$ can be continuously shrunk to any point $p \in \tilde{C}$ in such a way that the we have a family of $2$-intersecting curves throughout the process. In this paper, we generalize the result of Snoeyink and Hershberger to the setting of non-piercing curves. We show that given an arrangement of non-piercing curves $\Gamma$, and a fixed curve $\gamma\in \Gamma$, the arrangement can be swept by $\gamma$ so that the arrangement remains non-piercing throughout the process. We also give a shorter and simpler proof of the result of Snoeyink and Hershberger and cite applications of their result, where our result leads to a generalization.

翻译：设$\Gamma$为平面上有限条若尔当曲线构成的集合。对于任意曲线$\gamma \in \Gamma$，记$\tilde{\gamma}$为$\gamma$所围成的有界区域。称$\Gamma$为无穿刺族，若对任意两条曲线$\alpha, \beta \in \Gamma$，$\tilde{\alpha} \setminus \tilde{\beta}$均为连通区域。无穿刺曲线族推广了每条曲线对至多相交于两点的$2$-相交曲线族。Snoeyink与Hershberger（《曲线的扫描排列》，SoCG '89）证明：若给定$2$-相交曲线族$\mathcal{C}$及其中固定曲线$C\in\mathcal{C}$，则可通过$C$对该排列进行\emph{扫描}，即$C$可连续收缩至$\tilde{C}$内任意点$p$，且在此过程中始终保持$2$-相交曲线族性质。本文将该结果推广至无穿刺曲线情形：证明给定无穿刺曲线排列$\Gamma$及固定曲线$\gamma\in\Gamma$，可通过$\gamma$扫描该排列，使得整个过程中排列保持无穿刺性。同时给出Snoeyink与Hershberger结果更简短简洁的证明，并引用其结论的应用案例，表明我们的结果可推广相关应用。