Mathematische Zeitschrift

, Volume 279, Issue 1–2, pp 459–478 | Cite as

Semi-discrete constant mean curvature surfaces

  • Christian Müller


We study semi-discrete surfaces in three dimensional euclidean space which are defined on a parameter domain consisting of one smooth and one discrete parameter. More precisely, we consider only those surfaces which are glued together from individual developable surface strips. In particular we investigate minimal surfaces and constant mean curvature (CMC) surfaces with non vanishing mean curvature in the setting of Koenigs nets and Christoffel duality. We obtain incidence-geometric characterizations of the dualizability of Koenigs nets as well as for the Gauss image of CMC surfaces. We also consider isothermic semi-discrete CMC surfaces and a specific type of Cauchy problem in this regard.


Discrete differential geometry Semi-discrete surfaces  Minimal surfaces CMC surfaces 

Mathematics Subject Classification

39A12 53A05 53A10 53A40 



This research was supported in part by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’ through Grant I 706-N26 of the Austrian Science Fund (FWF).


  1. 1.
    Bobenko, A.I., Suris, Y.B.: Discrete Differential Geometry: Integrable Structure. American Math. Soc, Rhode Island (2008)CrossRefGoogle Scholar
  2. 2.
    Bobenko, A.I., Pottmann, H., Wallner, J.: A curvature theory for discrete surfaces based on mesh parallelity. Math Annalen 348, 1–24 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Burstall, F., Hertrich-Jeromin, U., Rossman, W., Santos, S.: Discrete surfaces of constant mean curvature. RIMS Kyokuroku Bessatsu (RIMS Proc.) 1880, 133–179 (2014)Google Scholar
  4. 4.
    Christoffel, E.B.: Ueber einige allgemeine Eigenschaften der Minimumsflächen. J. Reine Angew. Math. 67, 218–228 (1867)CrossRefzbMATHGoogle Scholar
  5. 5.
    Eschenburg, J.H., Jost, J.: Differentialgeometrie und Minimalflächen. Springer, Berlin (2007)zbMATHGoogle Scholar
  6. 6.
    Hertrich-Jeromin, U., Hoffmann, T., Pinkall, U.: A discrete version of the Darboux transform for isothermic surfaces. In: Discrete Integrable Geometry and Physics (Vienna, 1996), Oxford Lecture Ser. Math. Appl., vol. 16, pp. 59–81. Oxford Univ. Press, New York (1999)Google Scholar
  7. 7.
    Hoffmann, T.: Discrete cmc surfaces and discrete holomorphic maps. In: Discrete Integrable Geometry and Physics (Vienna, 1996), Oxford Lecture Ser. Math. Appl., vol. 16, pp. 97–112. Oxford Univ. Press, New York (1999)Google Scholar
  8. 8.
    Karpenkov, O., Wallner, J.: On offsets and curvatures for discrete and semidiscrete surfaces. Beitr. Algebra Geom. 55(1), 207–228 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Liu, Y., Xu, W., Wang, J., Zhu, L., Guo, B., Chen, F., Wang, G.: General planar quadrilateral mesh design using conjugate direction field. ACM Trans. Graph. 30, 140:1–140:10 (2011)Google Scholar
  10. 10.
    Müller, C.: On discrete constant mean curvature surfaces. Discrete Comput. Geom. 51(3), 516–538 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Müller, C., Wallner, J.: Semi-discrete isothermic surfaces. Results Math. 63(3–4), 1395–1407 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Polthier, K., Rossman, W.: Discrete constant mean curvature surfaces and their index. J. Reine Angew. Math. 549, 47–47 (2002)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Pottmann, H., Wallner, J.: Computational Line Geometry. Springer, Berlin (2001)zbMATHGoogle Scholar
  14. 14.
    Pottmann, H., Liu, Y., Wallner, J., Bobenko, A.I., Wang, W.: Geometry of multi-layer freeform structures for architecture. ACM Trans. Graphics 26(3), #65,1–11 (2007)Google Scholar
  15. 15.
    Pottmann, H., Schiftner, A., Bo, P., Schmiedhofer, H., Wang, W., Baldassini, N., Wallner, J.: Freeform surfaces from single curved panels. ACM Trans. Graphics 27(3), #76,1–10, proc. SIGGRAPH (2008)Google Scholar
  16. 16.
    Rossman, W., Yasumoto, M.: Weierstrass representation for semi-discrete minimal surfaces, and comparison of various discretized catenoids. J. Math-for-Ind 4B, 109–118 (2012)MathSciNetGoogle Scholar
  17. 17.
    Sauer, R.: Differenzengeometrie. Springer, Berlin (1970)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institute of Discrete Mathematics and GeometryVienna University of TechnologyViennaAustria

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