Abstract
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.
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References
Bobenko A., Pottmann H., Wallner J.: A curvature theory for discrete surfaces based on mesh parallelity. Math. Annalen 348, 1–24 (2010)
Bobenko, A., Suris, Yu.: Discrete differential geometry: integrable structure. Graduate Studies in Mathematics, vol. 98. American Mathematical Society, New York (2009)
Bobenko, A.I., Suris, Yu.: Discrete Koenigs nets and discrete isothermic surfaces. Int. Math. Res. Notices 1976–2012 (2009)
Pottmann, H.: Private communication (2007)
Pottmann H., Schiftner A., Bo P., Schmiedhofer H., Wang W., Baldassini N., Wallner J.: Freeform surfaces from single curved panels. ACM Trans. Graph. 27(3), article no. 76 (2008)
Pottmann, H., Wallner, J.: Computational line geometry. Mathematics and Visualization, Springer, Heidelberg (2001); 2nd Ed. (2010)
Sauer R.: Differenzengeometrie. Springer, Berlin (1970)
Springborn B., Schröder P., Ulrich P.: Conformal equivalence of triangle meshes. ACM Trans. Graph. 27(3), article no. 77 (2008)
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Müller, C., Wallner, J. Semi-Discrete Isothermic Surfaces. Results. Math. 63, 1395–1407 (2013). https://doi.org/10.1007/s00025-012-0292-4
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DOI: https://doi.org/10.1007/s00025-012-0292-4