Computing and Visualization in Science

, Volume 13, Issue 8, pp 377–396 | Cite as

Finding and classifying critical points of 2D vector fields: a cell-oriented approach using group theory

  • Felix EffenbergerEmail author
  • Daniel Weiskopf


We present a novel approach to finding critical points in cell-wise barycentrically or bilinearly interpolated vector fields on surfaces. The Poincaré index of the critical points is determined by investigating the qualitative behavior of 0-level sets of the interpolants of the vector field components in parameter space using precomputed combinatorial results, thus avoiding the computation of the Jacobian of the vector field at the critical points in order to determine its index. The locations of the critical points within a cell are determined analytically to achieve accurate results. This approach leads to a correct treatment of cases with two first-order critical points or one second-order critical point of bilinearly interpolated vector fields within one cell, which would be missed by examining the linearized field only. We show that for the considered interpolation schemes determining the index of a critical point can be seen as a coloring problem of cell edges. A complete classification of all possible colorings in terms of the types and number of critical points yielded by each coloring is given using computational group theory. We present an efficient algorithm that makes use of these precomputed classifications in order to find and classify critical points in a cell-by-cell fashion. Issues of numerical stability, construction of the topological skeleton, topological simplification, and the statistics of the different types of critical points are also discussed.


Vector field topology Interpolation Barycentric interpolation Linear interpolation Bilinear interpolation Level sets Higher-order singularities Computational group theory Colorings 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Supplementary material

791_2011_152_MOESM1_ESM.pdf (1.4 mb)
ESM 1 (PDF 1,470 kb)
791_2011_152_MOESM2_ESM.pdf (8.6 mb)
ESM 2 (PDF 8,807 kb) (12 kb)
ESM 3 (GAP 12 kb) (11 kb)
ESM 4 (GAP 11 kb) (7 kb)
ESM 5 (GAP 7 kb) (6 kb)
ESM 6 (GAP 7 kb)


  1. 1.
    Andronov A.A., Leontovich E.A., Gordon I.I., Maĭer A.G.: Qualitative Theory of Second-Order Dynamic Systems. Halsted Press, New York (1973)zbMATHGoogle Scholar
  2. 2.
    Andronov A.A., Pontryagin L.: Systêmes grosseris. Dokl. Akad. Nauk. SSSR. 14, 247–251 (1937)Google Scholar
  3. 3.
    Banks D.C., Linton S.A., Stockmeyer P.K.: Counting cases in substitope algorithms. IEEE Trans. Vis. Comput. Graph. 10(4), 371–384 (2004)CrossRefGoogle Scholar
  4. 4.
    Bendixson I.: Sur les courbes définies par des équations différentielles. Acta Math. 24, 1–88 (1901)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dumortier F., Llibre J., Artés J.C.: Qualitative Theory of Planar Differential Systems. Universitext. Springer, Berlin (2006)Google Scholar
  6. 6.
    Floater M.S., Hormann K.: Surface parameterization: a tutorial and survey. In: Dodgson, N.A., Floater, M.S., Sabin, M.A. (eds) Advances in Multiresolution for Geometric Modelling., pp. 157–186. Springer, Berlin (2005)CrossRefGoogle Scholar
  7. 7.
    The GAP Group: GAP—Groups, Algorithms, and Programming, Version 4.4. ( (2006)
  8. 8.
    Guckenheimer J., Holmes P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1990)Google Scholar
  9. 9.
    Helman, J.L., Hesselink, L.: Surface representations of two- and three-dimensional fluid flow topology. In: Proceedings of IEEE Conference on Visualization, pp. 6–13 (1990)Google Scholar
  10. 10.
    Helman J.L., Hesselink L.: Visualizing vector field topology in fluid flows. IEEE Comput. Graph. Appl. 11(3), 36–46 (1991)CrossRefGoogle Scholar
  11. 11.
    Huppert B.: Endliche Gruppen. I. Springer, Berlin (1967)zbMATHGoogle Scholar
  12. 12.
    IEEE Computer Society Standards Committee. Working group of the Microprocessor Standards Subcommittee, American National Standards Institute: IEEE standard for binary floating-point arithmetic. ANSI/IEEE Std 754-1985. IEEE Computer Society Press, Silver Spring, MD (1985)Google Scholar
  13. 13.
    Kälberer F., Nieser M., Polthier K.: Quadcover—surface parameterization using branched coverings. Comput. Graph. Forum. 26(3), 375–384 (2007)CrossRefGoogle Scholar
  14. 14.
    Laramee, R.S., Chen, G., Jankun-Kelly, M., Zhang, E., Thompson, D.: Bringing topology-based flow visualization to the application domain. In: Topology-based methods in visualization II, Math. Vis. Springer, Berlin, pp. 161–176 (2009)Google Scholar
  15. 15.
    Li, W.C., Vallet, B., Ray, N., Levy, B.: Representing higher-order singularities in vector fields on piecewise linear surfaces. IEEE Transactions on Visualization and Computer Graphics (Proceedings of IEEE Conference on Visualization) 12(5), 1315–1322 (2006)Google Scholar
  16. 16.
    Markus L.: Global structure of ordinary differential equations in the plane. Trans. Am. Math. Soc. 76, 127–148 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Nielson, G.M., Hamann, B.: The asymptotic decider: Resolving the ambiguity in marching cubes. In: Proceedings of IEEE Conference on Visualization, pp. 83–91 (1991)Google Scholar
  18. 18.
    O’Neill, B.: Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York. With applications to relativity (1983)Google Scholar
  19. 19.
    Perko L.: Differential Equations and Dynamical Systems. Springer, New York (1991)zbMATHGoogle Scholar
  20. 20.
    Poincaré H.: Mémoire sur les Courbes Définies par une Équation Différentielle. Éditions Jacques Gabay, Sceaux (1993)Google Scholar
  21. 21.
    Polthier, K., Preuß, E.: Identifying vector field singularities using a discrete Hodge decomposition. In: Visualization and mathematics III, Math. Vis., pp. 113–134. Springer, Berlin (2003)Google Scholar
  22. 22.
    Post F.H., Vrolijk B., Hauser H., Laramee R.S., Doleisch H.: The state of the art in flow visualisation: Feature extraction and tracking. Comput. Graph. Forum. 22(4), 775–792 (2003)CrossRefGoogle Scholar
  23. 23.
    Scheuermann, G., Hagen, H., Krüger, H., Menzel, M., Rockwood, A.: Visualization of higher order singularities in vector fields. In: Proceedings of IEEE Conference on Visualization, pp. 67–74 (1997)Google Scholar
  24. 24.
    Scheuermann G., Krüger H., Menzel M., Rockwood A.P.: Visualizing nonlinear vector field topology. IEEE Trans. Vis. Comput. Graph. 4(2), 109–116 (1998)CrossRefGoogle Scholar
  25. 25.
    Scheuermann G., Tricoche X.: Topological methods for flow visualization. In: Hansen, C.D., Johnson, C.R. (eds) The Visualization Handbook, pp. 341–356. Elsevier, Amsterdam (2005)CrossRefGoogle Scholar
  26. 26.
    Theisel, H.: Designing 2D vector fields of arbitrary topology. Computer Graphics Forum (Proceedings of Eurographics 2002) 21(3), 595–604 (2002)Google Scholar
  27. 27.
    Theisel, H., Róssl, C., Seidel, H.P.: Compression of 2D vector fields under guaranteed toplogy preservation. Computer Graphics Forum (Proceedings of Eurographics 2003) 22(3), 333–342 (2003)Google Scholar
  28. 28.
    Theisel, H., Weinkauf, T., Hege, H.C., Seidel, H.P.: Grid-independent detection of closed stream lines in 2d vector fields. In: Proceedings of Vision, Modeling and Visualization (VMV) 2004, pp. 421–428. Stanford, USA (2004)Google Scholar
  29. 29.
    Tricoche, X., Scheuermann, G., Hagen, H.: A topology simplification method for 2D vector fields. In: Proceedings of IEEE Conference on Visualization, pp. 359–366 (2000)Google Scholar
  30. 30.
    Wischgoll T., Scheuermann G.: Detection and visualization of closed streamlines in planar flows. IEEE Trans. Vis. Comput. Graph. 7(2), 165–172 (2001)CrossRefGoogle Scholar
  31. 31.
    Zhang E., Mischaikow K., Turk G.: Vector field design on surfaces. ACM Trans. Graph. 25(4), 1294–1326 (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Institut für Geometrie und TopologieUniversität StuttgartStuttgartGermany
  2. 2.Visualization Research Center (VISUS)Universität StuttgartStuttgartGermany

Personalised recommendations