Suryendu Dalal, Rahul Gangopadhyay, Rajiv Raman, Saurabh Ray
{"title":"平面内非穿透曲线的扫掠排列","authors":"Suryendu Dalal, Rahul Gangopadhyay, Rajiv Raman, Saurabh Ray","doi":"arxiv-2403.16474","DOIUrl":null,"url":null,"abstract":"Let $\\Gamma$ be a finite set of Jordan curves in the plane. For any curve\n$\\gamma \\in \\Gamma$, we denote the bounded region enclosed by $\\gamma$ as\n$\\tilde{\\gamma}$. We say that $\\Gamma$ is a non-piercing family if for any two\ncurves $\\alpha , \\beta \\in \\Gamma$, $\\tilde{\\alpha} \\setminus \\tilde{\\beta}$ is\na connected region. A non-piercing family of curves generalizes a family of\n$2$-intersecting curves in which each pair of curves intersect in at most two\npoints. Snoeyink and Hershberger (``Sweeping Arrangements of Curves'', SoCG\n'89) proved that if we are given a family $\\mathcal{C}$ of $2$-intersecting\ncurves and a fixed curve $C\\in\\mathcal{C}$, then the arrangement can be\n\\emph{swept} by $C$, i.e., $C$ can be continuously shrunk to any point $p \\in\n\\tilde{C}$ in such a way that the we have a family of $2$-intersecting curves\nthroughout the process. In this paper, we generalize the result of Snoeyink and\nHershberger to the setting of non-piercing curves. We show that given an\narrangement of non-piercing curves $\\Gamma$, and a fixed curve $\\gamma\\in\n\\Gamma$, the arrangement can be swept by $\\gamma$ so that the arrangement\nremains non-piercing throughout the process. We also give a shorter and simpler\nproof of the result of Snoeyink and Hershberger and cite applications of their\nresult, where our result leads to a generalization.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":"32 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Sweeping Arrangements of Non-Piercing Curves in Plane\",\"authors\":\"Suryendu Dalal, Rahul Gangopadhyay, Rajiv Raman, Saurabh Ray\",\"doi\":\"arxiv-2403.16474\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $\\\\Gamma$ be a finite set of Jordan curves in the plane. For any curve\\n$\\\\gamma \\\\in \\\\Gamma$, we denote the bounded region enclosed by $\\\\gamma$ as\\n$\\\\tilde{\\\\gamma}$. We say that $\\\\Gamma$ is a non-piercing family if for any two\\ncurves $\\\\alpha , \\\\beta \\\\in \\\\Gamma$, $\\\\tilde{\\\\alpha} \\\\setminus \\\\tilde{\\\\beta}$ is\\na connected region. A non-piercing family of curves generalizes a family of\\n$2$-intersecting curves in which each pair of curves intersect in at most two\\npoints. Snoeyink and Hershberger (``Sweeping Arrangements of Curves'', SoCG\\n'89) proved that if we are given a family $\\\\mathcal{C}$ of $2$-intersecting\\ncurves and a fixed curve $C\\\\in\\\\mathcal{C}$, then the arrangement can be\\n\\\\emph{swept} by $C$, i.e., $C$ can be continuously shrunk to any point $p \\\\in\\n\\\\tilde{C}$ in such a way that the we have a family of $2$-intersecting curves\\nthroughout the process. In this paper, we generalize the result of Snoeyink and\\nHershberger to the setting of non-piercing curves. We show that given an\\narrangement of non-piercing curves $\\\\Gamma$, and a fixed curve $\\\\gamma\\\\in\\n\\\\Gamma$, the arrangement can be swept by $\\\\gamma$ so that the arrangement\\nremains non-piercing throughout the process. We also give a shorter and simpler\\nproof of the result of Snoeyink and Hershberger and cite applications of their\\nresult, where our result leads to a generalization.\",\"PeriodicalId\":501570,\"journal\":{\"name\":\"arXiv - CS - Computational Geometry\",\"volume\":\"32 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-03-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - CS - Computational Geometry\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2403.16474\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - CS - Computational Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2403.16474","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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.