Results in Mathematics

, Volume 63, Issue 3–4, pp 1395–1407 | Cite as

Semi-Discrete Isothermic Surfaces

  • Christian Müller
  • Johannes Wallner


We study mappings of the form \({x : \mathbb{Z}\times\mathbb{R}\to\mathbb{R}^3}\) which can be seen as a limit case of purely discrete surfaces, or as a semi-discretization of smooth surfaces. In particular we discuss circular surfaces, isothermic surfaces, conformal mappings, and dualizability in the sense of Christoffel. We arrive at a semi-discrete version of Koenigs nets and show that in the setting of circular surfaces, isothermicity is the same as dualizability. We show that minimal surfaces constructed as a dual of a sphere have vanishing mean curvature in a certain well-defined sense, and we also give an incidence-geometric characterization of isothermic surfaces.

Mathematics Subject Classification (2000)

Primary 53A Secondary 


Semi-discrete surfaces Isothermic surfaces Minimal surfaces 


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Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.TU WienWienAustria
  2. 2.Institute of GeometryTU GrazGrazAustria

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