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On the Topology and Isotopic Meshing of Plane Algebraic Curves

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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' 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 \(\left\{ {f(\alpha ,y),\tfrac{{\partial f}}{{\partial y}}(\alpha ,y)} \right\}\) and simultaneously obtain numerous simple roots of f(α, y) = 0 on the α fiber. This significantly improves the efficiency of the lifting step because the authors are no longer required to compute the simple roots of f(α, y) = 0. After the topology is computed, a revised Newton'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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Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Hubei University of Science and Technology, Xianning, 437100, China

    Kai Jin

  2. Key Lab of Mathematics Mechanization Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Jinsan Cheng

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  1. Kai Jin
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  2. Jinsan Cheng
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Correspondence to Kai Jin.

Additional information

This research was supported by the National Natural Science Foundation of China under Grant No. 11471327, “The Research Funds for Beijing Universities” under Grant No. KM201910009001, and “The Research and Development Funds of Hubei University of Science and Technology” under Grant No. BK202024.

This paper was recommended for publication by Editor LI Hongbo.

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Jin, K., Cheng, J. On the Topology and Isotopic Meshing of Plane Algebraic Curves. J Syst Sci Complex 33, 230–260 (2020). https://doi.org/10.1007/s11424-020-8262-5

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  • DOI: https://doi.org/10.1007/s11424-020-8262-5

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