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On the Topology and Visualization of Plane Algebraic Curves

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Computer Algebra in Scientific Computing (CASC 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9301))

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Abstract

In this paper, we present a symbolic algorithm to compute the topology of a plane curve. The algorithm mainly involves resultant computations and real root isolation for univariate polynomials. The novelty of this paper is that we use a technique of interval polynomials to solve the system \(\big\{f(\alpha,\,y)=\frac{\partial f}{\partial y}(\alpha,\,y)=0\big\}\) and at the same time, get the simple roots of f(α, y) = 0 on the α fiber. It greatly improves the efficiency of the lifting step since we need not compute the simple roots of f(α, y) = 0 any more. After the topology is computed, we use a revised Newton’s method to compute the visualization of the plane algebraic curve. We ensure that the meshing is topologically correct. Many nontrivial examples show our implementation works well.

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Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Central China Normal University, Wuhan, China

    Kai Jin

  2. Key Lab of Mathematics Mechanization, AMSS, CAS, Beijing, China

    Jin-San Cheng & Xiao-Shan Gao

Authors
  1. Kai Jin
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  2. Jin-San Cheng
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  3. Xiao-Shan Gao
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Correspondence to Kai Jin .

Editor information

Editors and Affiliations

  1. Laboratory of Information Technologies, Joint Institute of Nuclear Research, Dubna, Russia

    Vladimir P. Gerdt

  2. Institute for Mathematics, Universität Kassel, Kassel, Germany

    Wolfram Koepf

  3. Institut für Mathematik, Universität Kassel, Kassel, Hessen, Germany

    Werner M. Seiler

  4. Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Jin, K., Cheng, JS., Gao, XS. (2015). On the Topology and Visualization of Plane Algebraic Curves. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2015. Lecture Notes in Computer Science(), vol 9301. Springer, Cham. https://doi.org/10.1007/978-3-319-24021-3_19

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  • DOI: https://doi.org/10.1007/978-3-319-24021-3_19

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-24020-6

  • Online ISBN: 978-3-319-24021-3

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