Re2l: An efficient output-sensitive algorithm for computing Boolean operations on circular-arc polygons and its applications

The boundaries of conic polygons consist of conic segments or second degree curves. The conic polygon has two degenerate or special cases: the linear polygon and the circular-arc polygon. The natural problem–Boolean operations on linear polygons–has been extensively studied. Surprisingly, (almost) n...

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Veröffentlicht in:	Computer aided design Jg. 83; S. 1 - 14
Hauptverfasser:	Wang, Zhi-Jie, Lin, Xiao, Fang, Mei-E, Yao, Bin, Peng, Yong, Guan, Haibing, Guo, Minyi
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Amsterdam Elsevier Ltd 01.02.2017 Elsevier BV
Schlagworte:	Algorithms Appendix points Boolean algebra Boolean operations Circular-arc polygons Circularity Complexity Intersections Polygons Related edges Segments Sequence lists Software reviews Boolean operations Circular-arc polygons Appendix points Sequence lists Related edges
ISSN:	0010-4485, 1879-2685
Online-Zugang:	Volltext
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Abstract	The boundaries of conic polygons consist of conic segments or second degree curves. The conic polygon has two degenerate or special cases: the linear polygon and the circular-arc polygon. The natural problem–Boolean operations on linear polygons–has been extensively studied. Surprisingly, (almost) no article focuses on the problem of Boolean operations on circular-arc polygons, yet this potentially has many applications. This implies that if there is a targeted solution for Boolean operations on circular-arc polygons, it should be favourable for potential users. In this article, we close the gap by devising a concise data structure and then developing a targeted algorithm called Re2l. Our method is simple, easy-to-implement, but without loss of efficiency. Given two circular-arc polygons with m and n edges respectively, our method runs in O(m+n+(l+k)logl) time, using O(m+n+k) space, where k is the number of intersections, and l is the number of related edges (defined in Section 5). Our algorithm has the power to approximate to linear complexity when k and l are small. The superiority of the proposed algorithm is also validated through empirical study. In particular, our method is of independent interest and we show that it can be easily extended to compute Boolean operations of other types of polygons. •We highlight the circular-arc polygon is one of special cases of the conic polygon, and Boolean operation on circular-arc polygons also has many applications.•We devise a concise and easy-to-operate data structure, and develop a targeted algorithm for Boolean operations on circular-arc polygons.•While this paper focuses on Boolean operations of circular-arc polygons, we show our techniques can be easily extended to compute Boolean operations of other types of polygons.•We provide the rigorous and detailed theoretical analysis for our algorithm.•We conduct extensive experiments to demonstrate the efficiency and effectiveness of our solution.
AbstractList	The boundaries of conic polygons consist of conic segments or second degree curves. The conic polygon has two degenerate or special cases: the linear polygon and the circular-arc polygon. The natural problem–Boolean operations on linear polygons–has been extensively studied. Surprisingly, (almost) no article focuses on the problem of Boolean operations on circular-arc polygons, yet this potentially has many applications. This implies that if there is a targeted solution for Boolean operations on circular-arc polygons, it should be favourable for potential users. In this article, we close the gap by devising a concise data structure and then developing a targeted algorithm called Re2l. Our method is simple, easy-to-implement, but without loss of efficiency. Given two circular-arc polygons with m and n edges respectively, our method runs in O(m+n+(l+k)logl) time, using O(m+n+k) space, where k is the number of intersections, and l is the number of related edges (defined in Section 5). Our algorithm has the power to approximate to linear complexity when k and l are small. The superiority of the proposed algorithm is also validated through empirical study. In particular, our method is of independent interest and we show that it can be easily extended to compute Boolean operations of other types of polygons. The boundaries of conic polygons consist of conic segments or second degree curves. The conic polygon has two degenerate or special cases: the linear polygon and the circular-arc polygon. The natural problem–Boolean operations on linear polygons–has been extensively studied. Surprisingly, (almost) no article focuses on the problem of Boolean operations on circular-arc polygons, yet this potentially has many applications. This implies that if there is a targeted solution for Boolean operations on circular-arc polygons, it should be favourable for potential users. In this article, we close the gap by devising a concise data structure and then developing a targeted algorithm called Re2l. Our method is simple, easy-to-implement, but without loss of efficiency. Given two circular-arc polygons with m and n edges respectively, our method runs in O(m+n+(l+k)logl) time, using O(m+n+k) space, where k is the number of intersections, and l is the number of related edges (defined in Section 5). Our algorithm has the power to approximate to linear complexity when k and l are small. The superiority of the proposed algorithm is also validated through empirical study. In particular, our method is of independent interest and we show that it can be easily extended to compute Boolean operations of other types of polygons. •We highlight the circular-arc polygon is one of special cases of the conic polygon, and Boolean operation on circular-arc polygons also has many applications.•We devise a concise and easy-to-operate data structure, and develop a targeted algorithm for Boolean operations on circular-arc polygons.•While this paper focuses on Boolean operations of circular-arc polygons, we show our techniques can be easily extended to compute Boolean operations of other types of polygons.•We provide the rigorous and detailed theoretical analysis for our algorithm.•We conduct extensive experiments to demonstrate the efficiency and effectiveness of our solution.
Author	Guan, Haibing Fang, Mei-E Yao, Bin Peng, Yong Wang, Zhi-Jie Guo, Minyi Lin, Xiao
Author_xml	– sequence: 1 givenname: Zhi-Jie surname: Wang fullname: Wang, Zhi-Jie email: cszhijwang@comp.polyu.edu.hk organization: Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai, China – sequence: 2 givenname: Xiao surname: Lin fullname: Lin, Xiao email: lin6008@126.com organization: Department of Science and Engineering, University of Shanghai for Science and Technology, Shanghai, China – sequence: 3 givenname: Mei-E surname: Fang fullname: Fang, Mei-E email: fme@hdu.edu.cn organization: School of Computer Science, Hangzhou Dianzi University, Hangzhou, China – sequence: 4 givenname: Bin surname: Yao fullname: Yao, Bin email: yaobin@cs.sjtu.edu.cn organization: Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai, China – sequence: 5 givenname: Yong surname: Peng fullname: Peng, Yong email: standy.peng@gmail.com organization: School of Computer Science, Hangzhou Dianzi University, Hangzhou, China – sequence: 6 givenname: Haibing surname: Guan fullname: Guan, Haibing email: hbguan@cs.sjtu.edu.cn organization: Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai, China – sequence: 7 givenname: Minyi surname: Guo fullname: Guo, Minyi email: guo-my@cs.sjtu.edu.cn organization: Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai, China
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Cites_doi	10.1109/TKDE.2014.2345402 10.1111/j.1467-8659.1987.tb00540.x 10.1016/0097-8493(89)90059-9 10.1016/j.cageo.2007.03.002 10.1145/274363.274364 10.1016/j.cag.2004.11.001 10.1145/1978802.1978811 10.1016/S0097-8493(00)00090-X 10.1016/0010-4485(95)00050-X 10.1016/j.cageo.2008.08.009 10.1109/TC.1979.1675432 10.1111/j.1467-8659.1989.tb00484.x 10.1145/381677.381691 10.1145/965141.563896 10.1007/s11390-009-9246-z 10.1145/360767.360802 10.1109/38.41469 10.1007/3-540-45749-6_19 10.1007/BF01994114 10.1145/130881.130894 10.1145/182.358439 10.1145/129902.129906 10.1109/SFCS.1976.16 10.1145/997817.997882 10.1007/s10707-015-0230-1 10.1145/1998196.1998224 10.1016/j.cad.2007.01.010
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Snippet	The boundaries of conic polygons consist of conic segments or second degree curves. The conic polygon has two degenerate or special cases: the linear polygon...
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SubjectTerms	Algorithms Appendix points Boolean algebra Boolean operations Circular-arc polygons Circularity Complexity Intersections Polygons Related edges Segments Sequence lists Software reviews
Title	Re2l: An efficient output-sensitive algorithm for computing Boolean operations on circular-arc polygons and its applications
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