Vector Field Second Order Derivative Approximation and Geometrical Characteristics

  • Michal Smolik
  • Vaclav Skala
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10404)


Vector field is mostly linearly approximated for the purpose of classification and description. This approximation gives us only basic information of the vector field. We will show how to approximate the vector field with second order derivatives, i.e. Hessian and Jacobian matrices. This approximation gives us much more detailed description of the vector field. Moreover, we will show the similarity of this approximation with conic section formula.


Vector field Critical point Geometry Conic section Hessian matrix 



The authors would like to thank their colleagues at the University of West Bohemia, Plzen, for their comments and suggestions, their valuable comments and hints provided. The research was supported by projects Czech Science Foundation (GACR) No. 17-05534S and partly by SGS 2016-013.


  1. 1.
    Agranovsky, A., Camp, D., Joy, K.I., Childs, H.: Subsampling-based compression and flow visualization. In: IS&T/SPIE Electronic Imaging, International Society for Optics and Photonics (2015)Google Scholar
  2. 2.
    Balduzzi, F., Bianchini, A., Maleci, R., Ferrara, G., Ferrari, L.: Critical issues in the CFD simulation of Darrieus wind turbines. Renew. Energy 85, 419–435 (2016)CrossRefGoogle Scholar
  3. 3.
    Benbourhim, M.N., Bouhamidi, A.: Approximation of vectors fields by thin plate splines with tension. J. Approx. Theory 136(2), 198–229 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cabrera, D.A.C., González-Casanova, P., Gout, C., Juárez, L.H., Reséndizd, L.R.: Vector field approximation using radial basis functions. J. Comput. Appl. Math. 240, 163–173 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Forsberg, A., Chen, J., Laidlaw, D.: Comparing 3D vector field visualization methods: a user study. IEEE Trans. Vis. Comput. Graph. 15(6), 1219–1226 (2009)CrossRefGoogle Scholar
  6. 6.
    Helman, J., Hesselink, L.: Representation and display of vector field topology in fluid flow data sets. IEEE Comput. 22(8), 27–36 (1989)CrossRefGoogle Scholar
  7. 7.
    Koch, S., Kasten, J., Wiebel, A., Scheuermann, G., Hlawitschka, M.: 2D Vector field approximation using linear neighborhoods. Vis. Comput. 32, 1563–1578 (2015)CrossRefGoogle Scholar
  8. 8.
    Laidlaw, D.H., Kirby, R.M., Jackson, C.D., Davidson, J.S., Miller, T.S., Da Silva, M., Warrenand, W.H., Tarr, M.J.: Comparing 2D vector field visualization methods: a user study. IEEE Trans. Vis. Comput. Graph. 11(1), 59–70 (2005)CrossRefGoogle Scholar
  9. 9.
    Lage, M., Petronetto, F., Paiva, A., Lopes, H., Lewiner, T., Tavares, G.: Vector field reconstruction from sparse samples with applications. In: 19th Brazilian Symposium on Computer Graphics and Image Processing, SIBGRAPI (2006)Google Scholar
  10. 10.
    de Leeuw, W., van Liere, R.: Collapsing flow topology using area metrics. In: Proceedings of IEEE Visualization 1999, pp. 349–354 (1999)Google Scholar
  11. 11.
    Lu, K., Chaudhuri, A., Lee, T.Y., Shen, H.W., Wong, P.C.: Exploring vector fields with distribution-based streamline analysis. PacificVis, pp. 257–264 (2013)Google Scholar
  12. 12.
    Peng, C., Teng, Y., Hwang, B., Guo, Z., Wang, L.P.: Implementation issues and benchmarking of lattice Boltzmann method for moving rigid particle simulations in a viscous flow. Comput. Math. Appl. 72(2), 349–374 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Philippou, P.A., Strickland, R.N.: Vector field analysis and synthesis using threedimensional phase portraits. Graph. Models Image Process. 59(6), 446–462 (1997)CrossRefGoogle Scholar
  14. 14.
    Scheuermann, G., Krüger, H., Menzel, M., Rockwood, A.: Visualizing non-linear vector field topology. IEEE Trans. Vis. Comput. Graph. 4(2), 109–116 (1998)CrossRefGoogle Scholar
  15. 15.
    Skraba, P., Rosen, P., Wang, B., Chen, G., Bhatia, H., Pascucci, V.: Critical point cancellation in 3D vector fields: robustness and discussion. IEEE Trans. Vis. Comput. Graph. (2016)Google Scholar
  16. 16.
    Skraba, P., Wang, B., Chen, G., Rosen, P.: 2D vector field simplification based on robustness. In: Pacific Visualization Symposium (PacificVis), IEEE, pp. 49–56 (2014)Google Scholar
  17. 17.
    Smolik, M., Skala, V.: Spherical RBF vector field interpolation: experimental study. SAMI 2017, pp. 431–434, Slovakia (2017)Google Scholar
  18. 18.
    Smolik, M., Skala, V.: Vector field interpolation with radial basis functions. SIGRAD 2016, pp. 15–21, Sweden (2016)Google Scholar
  19. 19.
    Weinkauf, T., Theisel, H., Shi, K., Hege, H.-C., Seidel, H.-P.: Extracting higher order critical points and topological simplification of 3D vector fields. In: Proceedings of IEEE Visualization 2005, pp. 559–566, Minneapolis, U.S.A. (2005)Google Scholar
  20. 20.
    Westermann, R., Johnson, C., Ertl, T.: Topology-preserving smoothing of vector fields. IEEE Trans.Vis. Comput. Graph 7(3), 222–229 (2001)CrossRefGoogle Scholar
  21. 21.
    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

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Computer Science and Engineering, Faculty of Applied SciencesUniversity of West BohemiaPilsenCzech Republic

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