Skip to main content
SpringerLink
Log in
Book cover

International Workshop on Computer Algebra in Scientific Computing

CASC 2015: Computer Algebra in Scientific Computing pp 245–259Cite as

On the Topology and Visualization of Plane Algebraic Curves

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9301))

Abstract

In this paper, we present a symbolic algorithm to compute the topology of a plane curve. The algorithm mainly involves resultant computations and real root isolation for univariate polynomials. The novelty of this paper is that we use a technique of interval polynomials to solve the system \(\big\{f(\alpha,\,y)=\frac{\partial f}{\partial y}(\alpha,\,y)=0\big\}\) and at the same time, get the simple roots of f(α, y) = 0 on the α fiber. It greatly improves the efficiency of the lifting step since we need not compute the simple roots of f(α, y) = 0 any more. After the topology is computed, we use a revised Newton’s method to compute the visualization of the plane algebraic curve. We ensure that the meshing is topologically correct. Many nontrivial examples show our implementation works well.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Akritas, A.G.: An implementation of Vincents Theorem. Numerische Mathematik 36, 53–62 (1980)

    Google Scholar 

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

    Google Scholar 

  3. Alcázar, J.G., Schicho, J., Sendra, J.R.: A delineablity-based method for computing critical sets of algebraic surfaces. J. Symb. Comp. 42, 678–691 (2007)

    Google Scholar 

  4. Arnon, D.S., McCallum, S.: A polynomial-time algorithm for the topological type of a real algebraic curve. J. Symb. Comp. 5, 213–236 (1998)

    Google Scholar 

  5. Arnon, D.S., Collins, G., McCallum, S.: Cylindrical algebraic decomposition, I: the Basic Algorithm. SIAM Journal on Computing 13(4), 865–877 (1984)

    Google Scholar 

  6. Arnon, D.S., Collins, G., McCallum, S.: Cylindrical algebraic decomposition, II: an adjacency algorithm for plane. SIAM Journal on Computing 13(4), 878–889 (1984)

    Google Scholar 

  7. Arnon, D.S., Collins, G.: Cylindrical algebraic decomposition, III: an adjacency algorithm for 3D space. J. Symb. Comp. 5(1,2), 163–187 (1988)

    Google Scholar 

  8. Basu, S., Pollack, R., Roy, M.F.: Algorithm in real algebraic geometry. Algorithms and Computation in Mathematics, vol. 10, 2nd edn. Springer, Berlin (2006)

    Google Scholar 

  9. Beltrán, C., Leykin, A.: Robust Certified Numerical Homotopy Tracking. Foundations of Computational Mathematics 13(2), 253–295 (2013)

    Google Scholar 

  10. Berberich, E., Emeliyanenko, P., Kobel, A., Sagraloff, M.: Arrangement computation for planar algebraic curves. In: Proc. SNC, pp. 88–99 (2011)

    Google Scholar 

  11. Burr, M., Choi, S., Galehouse, B., Yap, C.: Complete subdivision algorithms, II: Isotopic meshing of algebraic curves. In: Proc. ACM ISSAC, pp. 87–94 (2008)

    Google Scholar 

  12. Cheng, S.W., Dey, T.K., Ramos, A., Ray, T.: Sampling and meshing a surface with guaranteed topology and geometry. In: Proc. Symp. on CG, pp. 280–289 (2004)

    Google Scholar 

  13. Cheng, J.S., Gao, X.S., Li, J.: Topology determination and isolation for implicit plane curves. In: Proc. ACM Symposium on Applied Computing, pp. 1140–1141 (2009)

    Google Scholar 

  14. Cheng, J.S., Jin, K.: A generic position based method for real root isolation of zero-dimensional polynomial systems. J. Symb. Comp. 68, 204–224 (2015)

    Google Scholar 

  15. Cheng, J.S., Lazard, S., Peñaranda, L., Pouget, M., Rouillier, F., Tsigaridas, E.: On the topology of real algebraic plane curves. Mathematics in Computer Science 4, 113–117 (2010)

    Google Scholar 

  16. Collins, G., Akritas, A.: Polynomial real roots isolation using Descartes’ rule of signs. In: ISSAC, pp. 272–275 (1976)

    Google Scholar 

  17. Eigenwillig, A., Kerber, M.: Exact and efficient 2d-arrangements of arbitrary algebraic curves. In: Proc. 19th Annual ACM-SIAM Symposium on Discrete Algorithm (SODA 2008), San Francisco, USA, January 2008, pp. 122–131 (2008)

    Google Scholar 

  18. Eigenwillig, A., Kerber, M., Wolpert, N.: Fast and exact geometric analysis of real algebraic plane curves. In: Proc. ACM ISSAC 2007, pp. 151–158. ACM Press (2007)

    Google Scholar 

  19. Fulton, W.: Introduction to intersection theory in algebraic geometry. In: CBMS Regional Conference Series in Mathematics, vol. 54. Published for the Conference Board of the Mathematical Sciences, Washington, DC (1984)

    Google Scholar 

  20. Gao, X.S., Li, M.: Rational Quadratic Approximation to Real Algebraic Curves. Comp. Aid. Geom. Des. 21, 805–828 (2004)

    Google Scholar 

  21. Hong, H.: An Efficient Method for Analyzing the Topology of Plane Real Algebraic Curves. Mathematics and Computers in Simulation 42(4–6), 571–582 (1996)

    Google Scholar 

  22. Gomes, A.J.P., Morgado, J.F.M., Pereira, E.S.: A BSP-based algorithm for dimensionally nonhomogeneous planar implicit curves with topological guarantees. ACM Trans. Graph. 28(2) (2009)

    Google Scholar 

  23. Kerber, M.: Geometric algorithms for algebraic curves and surfaces. Ph.D. Thesis, Universität des Saarlandes, Germany (2009)

    Google Scholar 

  24. Kerber, M., Sagraloff, M.: A worst-case bound for topology computation of algebraic curves. J. Symb. Comp. 47, 239–258 (2012)

    Google Scholar 

  25. Kvasov, B.I.: Methods of Shape-Preserving Spline Approximation. World Scientific, Singapore (2000)

    Google Scholar 

  26. Labs, O.: A list of challenges for real algebraic plane curve visualization software In: Nolinear Computational Geometry. The IMA Volumes, vol. 151., pp. 137–164. Springer, New York (2010)

    Google Scholar 

  27. Lazard, D.: CAD and topology of semi-algebraic sets. Mathematics in Computer Science 4(1), 93–112 (2010)

    Google Scholar 

  28. Liang, C., Mourrain, B., Pavone, J.P.: Subdivision methods for the topology of 2d and 3d implicit curves. In: Computational Methods for Algebraic Spline Surfaces. Springer-Verlag (2006)

    Google Scholar 

  29. Lorensen, W.E., Cline, H.E.: Marching cubes: a high resolution 3d surface construction algorithm. In: Proc. SIGGRAPH 1987. ACM Press (1987)

    Google Scholar 

  30. González-Vega, L., Necula, I.: Efficient topology determination of implicitly defined algebraic plane curves. Comp. Aid. Geom. Des. 19, 719–743 (2002)

    Google Scholar 

  31. Martin, R., Shou, H., Voiculescu, I., Bowyer, A., Wang, G.: Comparison of interval methods for plotting algebraic curves. Comp. Aid. Geom. Des. 19, 553–587 (2002)

    Google Scholar 

  32. Mccallum, S., Collins, G.E.: Local box adjacency a lgorithms for cylindrical algebraic decompositions. J. Symb. Comp. 33, 321–342 (2002)

    Google Scholar 

  33. Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. Society for Industrial and Applied Mathematics, Philadelphia (2009)

    Google Scholar 

  34. Mourrain, B., Pion, S., Schmitt, S., Técourt, J.P., Tsigaridas, E.P., Wolpert, N.: Algebraic issues in computational geometry. In: Boissonnat, J.D., Teillaud, M. (eds.) Effective Computaional Geometry for Curves and Surfaces. Mathematics and Visualization, chapter 3. Springer, Berlin (2006)

    Google Scholar 

  35. Noakes, L., Kozera, R.: Cumulative chords, piecewise-quadratics and piecewise-cubics. In: Klette, R., et al. (eds.) Geometric Properties for Incomplete Data, pp. 59–75. Springer, Printed in the Netherlands (2006)

    Google Scholar 

  36. Plantinga, S., Vegter, G.: Isotopic meshing of implicit surfaces. Visual Computer 23, 45–58 (2007)

    Google Scholar 

  37. Ratschek, H.: Scci-hybrid method for 2d curve tracing. International Journal of Image and Graphics World Scientific Publishing Company 5(3), 447–479 (2005)

    Google Scholar 

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

    Google Scholar 

  39. Sakkalis, T.: The topological configuration of a real algebraic curve. Bull. Aust. Math. Soc. 43, 37–50 (1991)

    Google Scholar 

  40. Seidel, R., Wolpert, N., On the exact computation of the topology of real algebraic curves. In: Proceedings of Symposium on Computational Geometry, pp. 107–115 (2005)

    Google Scholar 

  41. Teissier, B.: Cycles évanescents, sections planes et conditions de Whitney. (french). Singularités à Cargèse. Astérisque 7 et 8, 285–362 (1973)

    Google Scholar 

  42. Li, T.Y.: Numerical solution of polynomial systems by homotopy continuation methods. Handbook of numerical analysis 11, 209–30 (2003)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Central China Normal University, Wuhan, China

    Kai Jin

  2. Key Lab of Mathematics Mechanization, AMSS, CAS, Beijing, China

    Jin-San Cheng & Xiao-Shan Gao

Authors
  1. Kai Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jin-San Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiao-Shan Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kai Jin .

Editor information

Editors and Affiliations

  1. Laboratory of Information Technologies, Joint Institute of Nuclear Research, Dubna, Russia

    Vladimir P. Gerdt

  2. Institute for Mathematics, Universität Kassel, Kassel, Germany

    Wolfram Koepf

  3. Institut für Mathematik, Universität Kassel, Kassel, Hessen, Germany

    Werner M. Seiler

  4. Novosibirsk, Russia

    Evgenii V. Vorozhtsov

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Jin, K., Cheng, JS., Gao, XS. (2015). On the Topology and Visualization of Plane Algebraic Curves. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2015. Lecture Notes in Computer Science(), vol 9301. Springer, Cham. https://doi.org/10.1007/978-3-319-24021-3_19

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-24021-3_19

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-24020-6

  • Online ISBN: 978-3-319-24021-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation