Sweeping Arrangements of Non-Piercing Curves in Plane
<|reference_start|>Sweeping Arrangements of Non-Piercing Curves in Plane: 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.<|reference_end|>
@article{dalal2024sweeping,
    title={Sweeping Arrangements of Non-Piercing Curves in Plane},
    author={Suryendu Dalal and Rahul Gangopadhyay and Rajiv Raman and Saurabh Ray},
    journal={arXiv preprint arXiv:2403.16474},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.16474},
    primaryClass={cs.CG}
}