On the Topology and Isotopic Meshing of Plane Algebraic Curves

JIN Kai · CHENG Jinsan

系统科学与复杂性(英文) ›› 2020, Vol. 33 ›› Issue (1) : 230-260.

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系统科学与复杂性(英文) ›› 2020, Vol. 33 ›› Issue (1) : 230-260. DOI: 10.1007/s11424-020-8262-5

On the Topology and Isotopic Meshing of Plane Algebraic Curves

    JIN Kai · CHENG Jinsan
作者信息 +

On the Topology and Isotopic Meshing of Plane Algebraic Curves

    JIN Kai · CHENG Jinsan
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Abstract

This paper presents a symbolic algorithm to compute the topology of a plane curve. This is a full version of the authors乫 CASC15 paper. The algorithm mainly involves resultant computations and real root isolation for univariate polynomials. Compared to other symbolic methods based on elimination techniques, the novelty of the proposed method is that the authors use a technique of interval polynomials to solve the system f(兛, y), 佪f 佪y (兛, y) and simultaneously obtain numerous simple roots of f(兛, y) = 0 on the 兛 fiber. This significantly improves the efficiency of the lifting step because the authors are no longer required to compute the simple roots of f(兛, y) = 0. After the topology is computed, a revised Newton乫s method is presented to compute an isotopic meshing of the plane algebraic curve. Though the approximation method is numerical, the authors can ensure that the proposed method is a certified one, and the meshing is topologically correct. Several nontrivial examples confirm that the proposed algorithm performs well.

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JIN Kai · CHENG Jinsan. On the Topology and Isotopic Meshing of Plane Algebraic Curves. 系统科学与复杂性(英文), 2020, 33(1): 230-260 https://doi.org/10.1007/s11424-020-8262-5
JIN Kai · CHENG Jinsan. On the Topology and Isotopic Meshing of Plane Algebraic Curves. Journal of Systems Science and Complexity, 2020, 33(1): 230-260 https://doi.org/10.1007/s11424-020-8262-5
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