research-article

On the topology of planar algebraic curves

Authors:

Sylvain Lazard,

Luis Pe�aranda,

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 Gr�bner 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.

References

[1]

L. Alberti, B. Mourrain, and J. Wintz. Topology and arrangement computation of semi-algebraic planar curves. Comput. Aided Geom. Des., 25(8):631--651, 2008.

Digital Library

[2]

D. Arnon and S. McCallum. A polynomial time algorithm for the topological type of a real algebraic curve. J. Symbolic Computation, 5:213--236, 1988.

Digital Library

[3]

D. S. Arnon, G. E. Collins, and S. McCallum. Cylindrical algebraic decomposition ii: An adjacency algorithm for the plane. SIAM J. Comput., 13(4):878--889, 1984.

Digital Library

[4]

S. Basu, R. Pollack, and M.-R. Roy. Algorithms in Real Algebraic Geometry, volume 10 of Algorithms and Computation in Mathematics. Springer-Verlag, 2nd edition, 2006.

Digital Library

[5]

R. Benedetti and J. Risler. Real Algebraic and Semi-algebraic Sets, Actualites Mathematiques. Hermann, 1990.

[6]

C. W. Brown. Improved projection for cylindrical algebraic decomposition. J. Symb. Comput., 32(5):447--465, 2001.

Digital Library

[7]

C. W. Brown. Contructing cylindrical algebraic decomposition of the plane quickly, 2002. Manuscript, http://www.cs.usna.edu/wcbrown/.

[8]

C. Burnikel, S. Funke, K. Mehlhorn, S. Schirra, and S. Schmitt. A separation bound for real algebraic expressions. In Proc. 9th Annual European Symposium on Algorithms, volume 2161 of LNCS, pages 254--265. Springer-Verlag, 2001.

Digital Library

[9]

M. Burr, S.W.Choi, B. Galehouse, and C. Yap. Complete subdivision algorithms, ii: Isotopic meshing of singular algebraic curves. In Proc. Intl. Symp. on Symbolic & Algebraic Computation (ISSAC 2008), 2008.

Digital Library

[10]

CGAL: Computational Geometry Algorithms Library. http://www.cgal.org.

[11]

M. Coste and M. F. Roy. Thom's lemma, the coding of real algebraic numbers and the computation of the topology of semi-algebraic sets. J. Symb. Comput., 5(1/2):121--129, 1988.

Digital Library

[12]

D. Cox, J. Little, and D. O'Shea. Using Algebraic Geometry. Number 185 in Graduate Texts in Mathematics. Springer, New York, 2nd edition, 2005.

[13]

D. I. Diochnos, I. Z. Emiris, and E. P. Tsigaridas. On the complexity of real solving bivariate systems. In C. W. Brown, editor, Proc. Int. Symp. Symbolic and Algebraic Computation, pages 127--134, Waterloo, Canada, 2007.

Digital Library

[14]

A. Eigenwillig, M. Kerber, and N. Wolpert. Fast and exact geometric analysis of real algebraic plane curves. In C. W. Brown, editor, Proc. Int. Symp. Symbolic and Algebraic Computation, pages 151--158, Waterloo, Canada, 2007. ACM.

Digital Library

[15]

A. Eigenwillig, L. Kettner, W. Krandick, K. Mehlhorn, S. Schmitt, and N. Wolpert. A Descartes Algorithm for Polynomials with Bit--Stream Coefficients. In V. Ganzha, E. Mayr, and E. Vorozhtsov, editors, CASC, volume 3718 of LNCS, pages 138--149. Springer, 2005.

Digital Library

[16]

A. Eigenwillig, V. Sharma, and C. K. Yap. Almost tight recursion tree bounds for the Descartes method. In Proc. Int. Symp. on Symbolic and Algebraic Computation, pages 71--78, New York, NY, USA, 2006. ACM Press.

Digital Library

[17]

A. Eigenwilling and M. Kerber. Exact and efficient 2d-arrangements of arbitrary algebraic curves. In Proc. 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA08), pages 122--131, San Francisco, USA, January 2008. ACM-SIAM, ACM/SIAM.

Digital Library

[18]

I. Z. Emiris, B. Mourrain, and E. P. Tsigaridas. Real Algebraic Numbers: Complexity Analysis and Experimentation. In P. Hertling, C. Hoffmann, W. Luther, and N. Revol, editors, Reliable Implementations of Real Number Algorithms: Theory and Practice, volume 5045 of LNCS, pages 57--82. Springer Verlag, 2008. also available in www.inria.fr/rrrt/rr-5897.html.

Digital Library

[19]

J.-C. Faug�re. A new efficient algorithm for computing gr�bner bases without reduction to zero f₅. In International Symposium on Symbolic and Algebraic Computation Symposium -- ISSAC 2002, Villeneuve d'Ascq, France, Jul 2002.

Digital Library

[20]

H. Feng. Decomposition and Computation of the Topology of Plane Real Algebraic Curves. Ph.d. thesis, The Royal Institute of Technology, Stockholm, 1992.

[21]

FGb -- A software for computing Gr�bner bases. J.-C. Faug�re. http://fgbrs.lip6.fr.

[22]

M. Giusti, G. Lecerf, and B. Salvy. A Gr�bner free alternative for solving polynomial systems. J. of Complexity, 17(1):154--211, 2001.

Digital Library

[23]

L. Gonz�lez-Vega and M. El Kahoui. An improved upper complexity bound for the topology computation of a real algebraic plane curve. J. Complexity, 12(4):527--544, 1996.

Digital Library

[24]

L. Gonz�lez-Vega, H. Lombardi, T. Recio, and M.-F. Roy. Sturm-Habicht Sequence. In Proc. Int. Symp. on Symbolic and Algebraic Computation, pages 136--146, 1989.

Digital Library

[25]

L. Gonz�lez-Vega and I. Necula. Efficient topology determination of implicitly defined algebraic plane curves. Computer Aided Geometric Design, 19(9), 2002.

Digital Library

[26]

G.-M. Greuel, G. Pfister, and H. Sch�nemann. Singular 3.0 -- a computer algebra system for polynomial computations. In M. Kerber and M. Kohlhase, editors, Symbolic computation and automated reasoning, The Calculemus-2000 Symposium, pages 227--233. A. K. Peters, Ltd., Natick, MA, USA, 2001.

Digital Library

[27]

H. Hong. An efficient method for analyzing the topology of plane real algebraic curves. Mathematics and Computers in Simulation, 42(4-6):571--582, 1996.

Digital Library

[28]

M. Kerber. Analysis of real algebraic plane curves. Master's thesis, MPII, 2006.

[29]

J. Keyser, K. Ouchi, and M. Rojas. The exact rational univariate representation for detecting degeneracies. In DIMACS Series in Discrete Mathematics and Theoretical Computer Science. AMS Press, 2005.

[30]

O. Labs. A list of challenges for real algebraic plane curve visualization software. Manuscript, 2008.

[31]

C. Li, S. Pion, and C. Yap. Recent progress in exact geometric computation. J. of Logic and Algebraic Programming, 64(1):85--111, 2004. Special issue on "Practical Development of Exact Real Number Computation".

[32]

S. McCallum and G. E. Collins. Local box adjacency algorithms for cylindrical algebraic decompositions. J. Symb. Comput., 33(3):321--342, 2002.

Digital Library

[33]

B. Mourrain, S. Pion, S. Schmitt, J.-P. T�court, E. P. Tsigaridas, and N. Wolpert. Algebraic issues in Computational Geometry. In J.-D. Boissonnat and M. Teillaud, editors, Effective Computational Geometry for Curves and Surfaces, Mathematics and Visualization, chapter 3. Springer, 2006.

[34]

B. Mourrain and P. Tr�buchet. Generalized normal forms and polynomial system solving. In Proc. Int. Symp. Symbolic and Algebraic Computation, pages 253--260, 2005.

Digital Library

[35]

F. Rouillier. Solving zero-dimensional systems through the rational univariate representation. J. of Applicable Algebra in Engineering, Communication and Computing, 9(5):433--461, 1999.

[36]

F. Rouillier and P. Zimmermann. Efficient isolation of polynomial real roots. J. of Computational and Applied Mathematics, 162(1):33--50, 2003.

Digital Library

[37]

RS -- A software for real solving of algebraic systems. F. Rouillier. http://fgbrs.lip6.fr.

[38]

T. Sakkalis. The topological configuration of a real algebraic curve. Bull. Austrl. Math. Soc., 43:37--50, 1991.

[39]

T. Sakkalis and R. Farouki. Singular points of algebraic curves. J. Symb. Comput., 9(4):405--421, 1990.

Digital Library

[40]

R. Seidel and N. Wolpert. On the exact computation of the topology of real algebraic curves. In Proc 21st ACM Symposium on Computational Geometry, pages 107--115, 2005.

Digital Library

[41]

A. Strzebonski. Cylindrical algebraic decomposition using validated numerics. J. Symb. Comput., 41(9):1021--1038, 2006.

[42]

B. Teissier. Cycles �vanescents, sections planes et conditions de Whitney. (french). In Singularit�s � Carg�se (Rencontre Singularit�s G�om. Anal., Inst. �tudes Sci., Carg�se, 1972), number 7--8 in Asterisque, pages 285--362. Soc. Math. France, Paris, 1973.

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
Show More Cited By

Index Terms

On the topology of planar algebraic curves

Recommendations

On the exact computation of the topology of real algebraic curves
SCG '05: Proceedings of the twenty-first annual symposium on Computational geometry

We consider the problem of computing a representation of the plane graph induced by one (or more) algebraic curves in the real plane. We make no assumptions about the curves, in particular we allow arbitrary singularities and arbitrary intersection. ...
Planar piecewise algebraic curves

An approach is described for piecing together segments of planar algebraic curves with derivative continuity. The application of piecewise algebraic curves to area modelling (the two-dimensional analogue of solid modelling) is discussed. A technique is ...
Planar Pythagorean-Hodograph B-Spline curves

We introduce a new class of planar Pythagorean-Hodograph (PH) B-Spline curves. They can be seen as a generalization of the well-known class of planar Pythagorean-Hodograph (PH) Bzier curves, presented by R. Farouki and T. Sakkalis in 1990, including the ...

Comments

Information & Contributors

Information

Published In

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.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 08 June 2009

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Conference

SoCG '09

Sponsor:

SoCG '09: 25th Annual Symposium on Computational Geometry

June 8 - 10, 2009

Aarhus, Denmark

Acceptance Rates

Overall Acceptance Rate 517 of 1,373 submissions, 38%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

12
Total Citations
View Citations
186
Total Downloads

Downloads (Last 12 months)6
Downloads (Last 6 weeks)1

Reflects downloads up to 17 Oct 2024

Other Metrics

View Author Metrics

Citations

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
https://dl.acm.org/doi/10.1145/2331684.2331710
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
https://dl.acm.org/doi/10.1007/s00373-011-1020-7
Cheng JLazard SPe�aranda 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
https://doi.org/10.1007/s11786-010-0044-3
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
https://dl.acm.org/doi/10.5555/3116258.3116373
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
https://doi.org/10.1007/s00454-009-9173-3
Show More Cited By

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents