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Mathematics in Computer Science

, Volume 4, Issue 1, pp 113–137 | Cite as

On the Topology of Real Algebraic Plane Curves

  • Jinsan Cheng
  • Sylvain Lazard
  • Luis Peñaranda
  • Marc Pouget
  • Fabrice Rouillier
  • Elias Tsigaridas
Article

Abstract

We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given coordinate system even if the curve is not in generic position. Previous methods based on the cylindrical algebraic decomposition use sub-resultant sequences and computations with polynomials with algebraic coefficients. A novelty of our approach is to replace these tools by Gröbner basis computations and isolation with rational univariate representations. This has the advantage of avoiding computations with polynomials with algebraic coefficients, even in non-generic positions. Our algorithm isolates critical points in boxes and computes a decomposition of the plane by rectangular boxes. This decomposition also induces a new approach for computing an arrangement of polylines isotopic to the input curve. We also present an analysis of the complexity of our algorithm. An implementation of our algorithm demonstrates its efficiency, in particular on high-degree non-generic curves.

Keywords

Singular Point Extreme Point Algebraic Curf Interval Arithmetic Algebraic Computation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Jinsan Cheng
    • 1
  • Sylvain Lazard
    • 2
  • Luis Peñaranda
    • 2
  • Marc Pouget
    • 2
  • Fabrice Rouillier
    • 3
    • 4
  • Elias Tsigaridas
    • 5
  1. 1.KLMM, Institute of Systems ScienceAMSS, Chinese Academy of SciencesBeijingChina
  2. 2.INRIA Nancy Grand Est, LORIA LaboratoryNancyFrance
  3. 3.INRIA Paris-RocquencourtParisFrance
  4. 4.LIP6 (Université Paris 6, CNRS)ParisFrance
  5. 5.Department of Computer ScienceAarhus UniversityAarhusDenmark

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