Automatic Generation of Riemann Surface Meshes

  • Matthias Nieser
  • Konstantin Poelke
  • Konrad Polthier
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6130)


Riemann surfaces naturally appear in the analysis of complex functions that are branched over the complex plane. However, they usually possess a complicated topology and are thus hard to understand. We present an algorithm for constructing Riemann surfaces as meshes in \({\mathbb R}^3\) from explicitly given branch points with corresponding branch indices. The constructed surfaces cover the complex plane by the canonical projection onto \({\mathbb R}^2\) and can therefore be considered as multivalued graphs over the plane – hence they provide a comprehensible visualization of the topological structure.

Complex functions are elegantly visualized using domain coloring on a subset of \({\mathbb C}\). By applying domain coloring to the automatically constructed Riemann surface models, we generalize this approach to deal with functions which cannot be entirely visualized in the complex plane.


Riemann Surface Complex Plane Branch Point Automatic Generation Height Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Yin, X., Jin, M., Gu, X.: Computing shortest cycles using universal covering space. Vis. Comput. 23(12), 999–1004 (2007)CrossRefGoogle Scholar
  2. 2.
    Kälberer, F., Nieser, M., Polthier, K.: Quadcover - surface parameterization using branched coverings. Comput. Graph. Forum. 26(3), 375–384 (2007)CrossRefGoogle Scholar
  3. 3.
    Trott, M.: Visualization of Riemann surfaces (2009), (retrieved December 8, 2009)
  4. 4.
    Trott, M.: Visualization of Riemann surfaces of algebraic functions. Mathematica in Education and Research 6, 15–36 (1997)Google Scholar
  5. 5.
    Farris, F.A.: Visualizing complex-valued functions in the plane, (retrieved December 8, 2009)
  6. 6.
    Pergler, M.: Newton’s method, Julia and Mandelbrot sets, and complex coloring, (retrieved December 8, 2009)
  7. 7.
    da Silva, E.L.: Reviews of functions of one complex variable graphical representation from software development for learning support, (retrieved on December 8, 2009)
  8. 8.
    Lundmark, H.: Visualizing complex analytic functions using domain coloring (2004), (retrieved on December 8, 2009)
  9. 9.
    Hlavacek, J.: Complex domain coloring, (retrieved on December 8, 2009)
  10. 10.
    Poelke, K., Polthier, K.: Lifted domain coloring. Computer Graphics Forum 28(3), 735–742 (2009)CrossRefGoogle Scholar
  11. 11.
    Erickson, J., Whittlesey, K.: Greedy optimal homotopy and homology generators. In: SODA, pp. 1038–1046 (2005)Google Scholar
  12. 12.
    Farkas, H.: Riemann Surfaces. Springer, New York (1980)zbMATHGoogle Scholar
  13. 13.
    Lamotke, K.: Riemannsche Flächen. Springer, Berlin (2005)zbMATHGoogle Scholar
  14. 14.
    Forster, O.: Lectures on Riemann Surfaces, 4th edn. Graduate Texts in Mathematics. Springer, Heidelberg (1999)Google Scholar
  15. 15.
    Needham, T.: Visual Complex Analysis. Oxford University Press, Oxford (2000)Google Scholar
  16. 16.
    Kälberer, F., Nieser, M., Polthier, K.: Stripe parameterization of tubular surfaces. In: Topology-Based Methods in Visualization III. Mathematics and Visualization. Springer, Heidelberg (to appear 2010)Google Scholar
  17. 17.
    Riemann, B.: Grundlagen für eine allgemeine theorie der functionen einer veränderlichen complexen größe (1851)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Matthias Nieser
    • 1
  • Konstantin Poelke
    • 1
  • Konrad Polthier
    • 1
  1. 1.Freie Universität BerlinGermany

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