A discrete version of the Darboux transform for isothermic surfaces

Details A discrete version of the Darboux transform for isothermic surfaces A discrete version of the Darboux transform for isothermic surfaces

1996-11-25

arxiv.org/abs/dg-ga/9611009v1

Mathematics

Differential Geometry

30 pages, LaTeX, a version with high quality figures is available at http://www-sfb288.math.tu-berlin.de/preprints.html

Scientific paper

We study Christoffel and Darboux transforms of discrete isothermic nets in 4-dimensional Euclidean space: definitions and basic properties are derived. Analogies with the smooth case are discussed and a definition for discrete Ribaucour congruences is given. Surfaces of constant mean curvature are special among all isothermic surfaces: they can be characterized by the fact that their parallel constant mean curvature surfaces are Christoffel and Darboux transforms at the same time. This characterization is used to define discrete nets of constant mean curvature. Basic properties of discrete nets of constant mean curvature are derived.

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