Advertisement

Journal of Zhejiang University-SCIENCE A

, Volume 9, Issue 12, pp 1685–1693 | Cite as

Computing the topology of an arrangement of implicitly defined real algebraic plane curves

  • Jorge Caravantes
  • Laureano Gonzalez-Vega
Article

Abstract

We introduce a new algebraic approach dealing with the problem of computing the topology of an arrangement of a finite set of real algebraic plane curves presented implicitly. The main achievement of the presented method is a complete avoidance of irrational numbers that appear when using the sweeping method in the classical way for solving the problem at hand. Therefore, it is worth mentioning that the efficiency of the proposed method is only assured for low-degree curves.

Key words

Topology computation Real plane curves Sweeping method 

CLC number

TP391.72 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Basu, S., Pollack, R., Roy, M.F., 2006. Algorithms in Real Algebraic Geometry. Algorithms and Computation in Mathematics, Vol. 10. Springer.Google Scholar
  2. Berberich, E., Eigenwillig, A., Hemmer, M., Hert, S., Mehlhorn, K., Schömer, E., 2002. A computational basis for conic arcs and Boolean operations on conic polygons. LNCS, 2461:174–186.Google Scholar
  3. Caravantes, J., Gonzalez-Vega, L., 2007a. Computing the topology of an arrangement of quartics. LNCS, 4647: 104–120.Google Scholar
  4. Caravantes, J., Gonzalez-Vega, L., 2007b. Improving the topology computation of an arrangement of cubics. Comput. Geometr.: Theory Appl., 41(3):206–218.MathSciNetCrossRefMATHGoogle Scholar
  5. Cheng, J.S., Lazard, S., Penaranda, L., Pouget, M., Rouillier, F., Tsigaridas, E., 2008. On the Topology of Planar Algebraic Curves. Proc. 24th European Workshop on Computational Geometry, p.213–216.Google Scholar
  6. Eigenwillig, A., Kettner, L., Schömer, E., Wolpert, N., 2006. Exact, efficient and complete arrangement computation for cubic curves. Comput. Geometr., 35(1–2):36–73. [doi:10.1016/j.comgeo.2005.10.003]MathSciNetCrossRefMATHGoogle Scholar
  7. Eigenwillig, A., Kerber, M., Wolpert, N., 2007. Fast and Exact Geometric Analysis of Real Algebraic Plane Curves. Proc. Int. Symp. on Symbolic and Algebraic Computation. ACM Press, p.151–158.Google Scholar
  8. Flato, E., Halperin, D., Hanniel, I., Nechushtan, O., Ezra, E., 2000. The design and implementation of planar maps in CGAL. ACM J. Exp. Algor., 5:1–23 [doi:10.1145/351827.384255]MathSciNetMATHGoogle Scholar
  9. Gonzalez-Vega, L., El Kahoui, M., 1996. An improved upper complexity bound for the topology computation of a real algebraic plane curve. J. Compl., 12(4):527–544. [doi:10.1006/jcom.1996.0032]MathSciNetCrossRefMATHGoogle Scholar
  10. Gonzalez-Vega, L., Necula, I., 2002. Efficient topology determination of implicitly defined algebraic plane curves. Comput. Aided Geometr. Des., 19(9):719–743. [doi:10.1016/S0167-8396(02)00167-X]MathSciNetCrossRefMATHGoogle Scholar
  11. Hong, H., 1996. An efficient method for analyzing the topology of plane real algebraic curves. Math. Comput. Simul., 42(4–6):571–582. [doi:10.1016/S0378-4754(96)00034-1]MathSciNetCrossRefMATHGoogle Scholar
  12. Li, Y.B., 2006. A new approach for constructing subresultants. Appl. Math. Comput., 183:471–476. [doi:10.1016/j.amc.2006.05.120]MathSciNetMATHGoogle Scholar
  13. Liang, C., Mourrain, B., Pavone, J.P., 2007. Subdivision methods for the topology of 2D and 3D implicit curves. Geometr. Model. Algebr. Geometr., p.199–214.Google Scholar
  14. Mehlhorn, K., Noher, S., 1999. LEDA: A Platform for Combinatorial and Geometric Computing. Cambridge University Press.Google Scholar
  15. Sakkalis, T., 1991. The topological configuration of a real algebraic curve. Bull. Aust. Math. Soc., 43:37–50.MathSciNetCrossRefMATHGoogle Scholar
  16. Sakkalis, T., Farouki, R., 1990. Singular points of algebraic curves. J. Symb. Comput., 9(4):405–421. [doi:10.1016/S0747-7171(08)80019-3]MathSciNetCrossRefMATHGoogle Scholar
  17. Seel, M., 2001. Implementation of Planar Nef Polyhedra. Report MPI-I-2001-1-003, Max-Planck-Institut für Informatik.Google Scholar
  18. Seidel, R., Wolpert, N., 2005. On the Exact Computation of the Topology of Real Algebraic Curves (Exploiting a Little More Geometry and a Little Less Algebra). Proc. 21st Annual ACM Symp. on Computational Geometry. ACM Press, p.107–115.Google Scholar
  19. Wein, R., 2002. High-level filtering for arrangements of conic arcs. LNCS, 2461:884–895.Google Scholar

Copyright information

© Zhejiang University and Springer-Verlag GmbH 2008

Authors and Affiliations

  1. 1.Department of Mathematics, Statistics and ComputationUniversity of Cantabria, SantanderCantabriaSpain

Personalised recommendations