research-article

On the topology of planar algebraic curves

Authors:

Sylvain Lazard,

Luis Peñaranda,

Fabrice Rouillier,

Elias TsigaridasAuthors Info & Claims

SCG '09: Proceedings of the twenty-fifth annual symposium on Computational geometry

Pages 361 - 370

https://doi.org/10.1145/1542362.1542424

Published: 08 June 2009 Publication History

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 (CAD) 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.

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Blažková EŠír Z(2014)Exploiting the Implicit Support Function for a Topologically Accurate Approximation of Algebraic CurvesMathematical Methods for Curves and Surfaces10.1007/978-3-642-54382-1_4(49-67)Online publication date: 2014
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On the topology of planar algebraic curves

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cover image ACM Conferences

SCG '09: Proceedings of the twenty-fifth annual symposium on Computational geometry

June 2009

426 pages

ISBN:9781605585017

DOI:10.1145/1542362

Program Chairs:
John Hershberger
Mentor Graphics Corp.
,
Efi Fogel
Tel Aviv University

Copyright © 2009 ACM.

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Published: 08 June 2009

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SoCG '09: 25th Annual Symposium on Computational Geometry

June 8 - 10, 2009

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Cited By

Blažková EŠír Z(2014)Identifying and approximating monotonous segments of algebraic curves using support function representationComputer Aided Geometric Design10.1016/j.cagd.2014.05.00631:7(358-372)Online publication date: 1-Oct-2014
https://dl.acm.org/doi/10.1016/j.cagd.2014.05.006
Blažková EŠír Z(2014)Exploiting the Implicit Support Function for a Topologically Accurate Approximation of Algebraic CurvesMathematical Methods for Curves and Surfaces10.1007/978-3-642-54382-1_4(49-67)Online publication date: 2014
https://doi.org/10.1007/978-3-642-54382-1_4
Pan VTsigaridas EMonagan MCooperman GGiesbrecht M(2013)On the boolean complexity of real root refinementProceedings of the 38th International Symposium on Symbolic and Algebraic Computation10.1145/2465506.2465938(299-306)Online publication date: 26-Jun-2013
https://dl.acm.org/doi/10.1145/2465506.2465938
Emeliyanenko PSagraloff Mvan der Hoeven Jvan Hoeij M(2012)On the complexity of solving a bivariate polynomial systemProceedings of the 37th International Symposium on Symbolic and Algebraic Computation10.1145/2442829.2442854(154-161)Online publication date: 22-Jul-2012
https://dl.acm.org/doi/10.1145/2442829.2442854
Kamath NVoiculescu IYap CKotsireas I(2012)Empirical study of an evaluation-based subdivision algorithm for complex root isolationProceedings of the 2011 International Workshop on Symbolic-Numeric Computation10.1145/2331684.2331710(155-164)Online publication date: 7-Jun-2012
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Islam MPoteaux A(2011)Connectivity queries on curves in RnACM Communications in Computer Algebra10.1145/2016567.201658445:1/2(117-118)Online publication date: 25-Jul-2011
https://dl.acm.org/doi/10.1145/2016567.2016584
Kerber MSagraloff M(2011)A Note on the Complexity of Real Algebraic HypersurfacesGraphs and Combinatorics10.1007/s00373-011-1020-727:3(419-430)Online publication date: 1-May-2011
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Cheng JLazard SPeñaranda LPouget MRouillier FTsigaridas E(2010)On the Topology of Real Algebraic Plane CurvesMathematics in Computer Science10.1007/s11786-010-0044-34:1(113-137)Online publication date: 9-Oct-2010
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Everett HLazard DLazard SSafey El Din M(2009)The Voronoi Diagram of Three LinesDiscrete & Computational Geometry10.5555/3116258.311637342:1(94-130)Online publication date: 1-Jul-2009
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Everett HLazard DLazard SSafey El Din M(2009)The Voronoi Diagram of Three LinesDiscrete & Computational Geometry10.1007/s00454-009-9173-342:1(94-130)Online publication date: 28-Apr-2009
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