Semi-Discrete isothermic surfaces

C. Müller and J. Wallner

Abstract:

We study mappings of the form x : Z × R -> R3 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 semidiscrete 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.

Bibtex:

@article{mueller-2013-sd,
        author = {Christian M{\"u}ller and Johannes Wallner},
        title = {Semi-{D}iscrete {I}sothermic {S}urfaces},
	journal = {Results Math.},
	volume = {63},
	year = {2013},
	number = {3-4},
	pages = {1395--1407},
        doi = {http://dx.doi.org/10.1007/s00025-012-0292-4},
        url = {http://www.geometrie.tuwien.ac.at/cmueller/conformal.pdf,
}

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