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The Visual Computer

, Volume 31, Issue 2, pp 141–154 | Cite as

Visualisation of complex functions on Riemann sphere

  • Miroslava ValíkováEmail author
  • Pavel Chalmovianský
Original Article

Abstract

The notion of a multi-valued function is frequent in complex analysis and related fields. A graph of such a function helps to inspect the function, however, the methods working with single-valued functions can not be applied directly. To visualize such a type of function, its Riemann surface is often used as a domain of the function. On such a surface, a multi-valued function behaves like a single-valued function. In our paper, we give a quick overview of the proposed method of visualization of a single-valued complex function over its Riemann sphere. Then, we pass to the adaptation of this method on the visualization of a multi-valued complex function. Our method uses absolute value and argument of the function to create the graph in 3D space over the Riemann sphere. Such a graph provides an overview of the function behavior over its whole domain, the amount and the position of its branch points, as well as poles and zeros and their multiplicity. We have also created an algorithm for adaptive grid which provides higher density of vertices in areas with higher curvature of the graph. The algorithm eliminates the alias in places where the branches are joined together.

Keywords

Riemann surfaces Multi-valued function Visualization Singular points 

Notes

Acknowledgments

The authors were kindly supported by the projects VEGA 1/0330/13 and UK/465/2013. The authors thank the anonymous reviewers for valuable comments which helped to improve the paper considerably.

References

  1. 1.
    Olver, F.W.J. and National Institute of Standards and Technology (U.S.): NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010). http://dlmf.nist.gov. Accessed 18 Feb 2014
  2. 2.
    Yin, X., Jin, M., Gu, X.: Computing Shortest Cycles Using Universal Covering Space. CAD/Graphics (2007)Google Scholar
  3. 3.
    Chen, C.-C., Lin, C.-S.: Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces. Comm. Pure Appl. Math. 55, 728–771 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Kälberer, F., Nieser, M., Polthier, K., Quadcover- surface parameterization using branched coverings. Comput. Graph. Forum (2007)Google Scholar
  5. 5.
    Bátorová, M. Valíková, M., Chalmovianský, P.: Desingularization of ADE Singularities via Deformation. In: Spring conference on Computer Graphics (2013), ACM Digital Library (to appear)Google Scholar
  6. 6.
    Greuel, G.M., Lossen, Ch., Shustin, E.: Introduction to Singularities and Deformations. Springer, New York (2007)zbMATHGoogle Scholar
  7. 7.
    Bobenko, A.I., Klein, Ch.: Computational Approach to Riemann Surfaces. Springer, Berlin (2011)CrossRefzbMATHGoogle Scholar
  8. 8.
    Farris, F.A.: Visualizing Complex-Valued Functions in the Plane. http://www.maa.org/pubs/amm_complements/complex.html. Accessed 18 Feb 2014
  9. 9.
    Lundmark, H.: Visualizing complex analytic functions using domain coloring (2004). http://www.mai.liu.se/halun/complex/. Accessed 18 Feb 2014
  10. 10.
    Poelke, K., Polthier, K.: Lifted domain coloring. Comput. Graph. Forum 28(3), 735–742 (2009)CrossRefGoogle Scholar
  11. 11.
    Poelke, K., Polthier, K.: Exploring Complex Functions Using Domain Coloring. (2012). http://news.cnet.com/2300-11386_3-10011177-6.html. Accessed 18 Feb 2014
  12. 12.
    Poelke, K., Polthier, K.: Domain coloring of complex functions: an implementation-iriented introduction. IEEE Comput. Graph. Appl. 32(5), 90–97 (2012)CrossRefGoogle Scholar
  13. 13.
    Wegert, E.: Visual Complex Functions: An Introduction with Phase Portraits. Springer, Berlin (2012)Google Scholar
  14. 14.
    Trott, M.: Riemann Surfaces over the Riemann Sphere (2009). http://www.mathematica-journal.com/issue/v8i4/columns/trott/contents/RiemannIId_2.html. Accessed 18 Feb 2014
  15. 15.
    Nieser, M., Poelke, K., Polthier, K.: Automatic generation of Riemann surface meshes. In: GMP’10 Proceedings of the 6th International Conference on Advances in Geometric Modeling and Processing (2010), pp. 161–178Google Scholar
  16. 16.
    Kranich, S.: Real-time Visualization of Geometric Singularities. Masters thesis (2012)Google Scholar
  17. 17.
    Lang, S.: Complex Analysis, 4th edn. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  18. 18.
    Valíková, M., Chalmovianský, P.: Visualization of the multi-valued complex function using the Riemann surface. In: Proceedings of Symposium on Computer Geometry SCG 2012 (2012)Google Scholar
  19. 19.
    Blender, http://www.blender.org/. Accessed 18 Feb 2014
  20. 20.
    Python, http://www.python.org/. Accessed 18 Feb 2014
  21. 21.
    Jones, G.A., Singerman, D.: Complex Functions: An Algebraic and Geometric Viewpoint. Press Syndicate of the University of Cambridge, Great Britain (1987)CrossRefzbMATHGoogle Scholar
  22. 22.
    Hansen, G.A., Douglass, R.W., Zardecki, A.: Mesh Enhancement: Selected Elliptic Methods. Foundations and Applications. World Scientific Publishing Company, Singapore (2005)CrossRefGoogle Scholar
  23. 23.
    Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps: Classification of Critical Points, Caustics and Wave Fronts (reprint of the 1985 edition). Birkhäuser Boston, Inc (2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Faculty of Mathematics, Physics and InformaticsComenius UniversityBratislavaSlovakia

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