Visualizing Arcs of Implicit Algebraic Curves, Exactly and Fast

  • Pavel Emeliyanenko
  • Eric Berberich
  • Michael Sagraloff
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5875)


Given a Cylindrical Algebraic Decomposition [2] of an implicitly defined algebraic curve, visualizing distinct curve arcs is not as easy as it stands because, despite the absence of singularities in the interior, the arcs can pass arbitrary close to each other. We present an algorithm to visualize distinct arcs of algebraic curves efficiently and precise (at a given resolution), irrespective of how close to each other they actually pass. Our hybrid method inherits the ideas of subdivision and curve-tracking methods. With an adaptive mixed-precision model we can render the majority of curves using machine arithmetic without sacrificing the exactness of the final result. The correctness and applicability of our algorithm is borne out by the success of our web-demo presented in [11].


Algebraic curves geometric computing curve rendering visualization exact computation 


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  1. 1.
    Alberti, L., Mourrain, B.: Visualisation of Implicit Algebraic Curves. In: Pacific Conference on Computer Graphics and Applications, pp. 303–312 (2007)Google Scholar
  2. 2.
    Arnon, D.S., Collins, G.E., McCallum, S.: Cylindrical algebraic decomposition I: the basic algorithm. SIAM J. Comput. 13, 865–877 (1984)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Berberich, E., Emeliyanenko, P.: Cgal’s Curved Kernel via Analysis. Technical Report ACS-TR-123203-04, Algorithms for Complex Shapes (2008)Google Scholar
  4. 4.
    Berberich, E., Kerber, M.: Exact Arrangements on Tori and Dupin Cyclides. In: Haines, E., McGuire, M. (eds.) SPM 2008, pp. 59–66. ACM, Stony Brook (2008)CrossRefGoogle Scholar
  5. 5.
    Burr, M., Choi, S.W., Galehouse, B., Yap, C.K.: Complete subdivision algorithms, II: isotopic meshing of singular algebraic curves. In: ISSAC 2008, pp. 87–94. ACM, New York (2008)CrossRefGoogle Scholar
  6. 6.
    Chandler, R.: A tracking algorithm for implicitly defined curves. IEEE Computer Graphics and Applications 8 (1988)Google Scholar
  7. 7.
    Cheng, J., Lazard, S., Peñaranda, L., Pouget, M., Rouillier, F., Tsigaridas, E.: On the topology of planar algebraic curves. In: SCG 2009, pp. 361–370. ACM, New York (2009)CrossRefGoogle Scholar
  8. 8.
    Eigenwillig, A., Kerber, M., Wolpert, N.: Fast and exact geometric analysis of real algebraic plane curves. In: ISSAC 2007, pp. 151–158. ACM, New York (2007)CrossRefGoogle Scholar
  9. 9.
    Elber, G., Kim, M.-S.: Geometric constraint solver using multivariate rational spline functions. In: SMA 2001, pp. 1–10. ACM, New York (2001)CrossRefGoogle Scholar
  10. 10.
    Emeliyanenko, P.: Visualization of Points and Segments of Real Algebraic Plane Curves. Master’s thesis, Universität des Saarlandes (2007)Google Scholar
  11. 11.
    Emeliyanenko, P., Kerber, M.: Visualizing and exploring planar algebraic arrangements: a web application. In: SCG 2008, pp. 224–225. ACM, New York (2008)CrossRefGoogle Scholar
  12. 12.
    Huahao Shou, I.V., Martin, R., et al.: Affine arithmetic in matrix form for polynomial evaluation and algebraic curve drawing. Progress in Natural Science 12(1), 77–81 (2002)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Martin, R., Shou, H., Voiculescu, I., Bowyer, A., Wang, G.: Comparison of interval methods for plotting algebraic curves. Comput. Aided Geom. Des. 19, 553–587 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Mehlhorn, K., Sagraloff, M.: Isolating Real Roots of Real Polynomials. In: ISSAC 2009, pp. 247–254. ACM, New York (2009)CrossRefGoogle Scholar
  15. 15.
    Messine, F.: Extensions of Affine Arithmetic: Application to Unconstrained Global Optimization. Journal of Universal Computer Science 8, 992–1015 (2002)MathSciNetGoogle Scholar
  16. 16.
    Möller, T., Yagel, R.: Efficient Rasterization of Implicit Functions. Tech. rep., Department of Computer and Information Science, Ohio State University (1995)Google Scholar
  17. 17.
    Morgado, J., Gomes, A.: A Derivative-Free Tracking Algorithm for Implicit Curves with Singularities. In: ICCSA, pp. 221–228 (2004)Google Scholar
  18. 18.
    Plantinga, S., Vegter, G.: Isotopic approximation of implicit curves and surfaces. In: SGP 2004, pp. 245–254. ACM, New York (2004)CrossRefGoogle Scholar
  19. 19.
    Ratschek, H., Rokne, J.G.: SCCI-hybrid Methods for 2d Curve Tracing. Int. J. Image Graphics 5, 447–480 (2005)CrossRefGoogle Scholar
  20. 20.
    Seidel, R., Wolpert, N.: On the exact computation of the topology of real algebraic curves. In: SCG 2005, pp. 107–115. ACM, New York (2005)CrossRefGoogle Scholar
  21. 21.
    Yu, Z.S., Cai, Y.Z., Oh, M.J., et al.: An Efficient Method for Tracing Planar Implicit Curves. Journal of Zhejiang University - Science A 7, 1115–1123 (2006)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Pavel Emeliyanenko
    • 1
  • Eric Berberich
    • 2
  • Michael Sagraloff
    • 1
  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany
  2. 2.School of Computer ScienceTel-Aviv UniversityTel-AvivIsrael

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