This is a Darboux transform of a spherical discrete isothermic net
Smooth surfaces of constant mean curvature 1 in hyperbolic space can be characterized by the fact that a suitable Darboux transform (by means of the conformal Gauss map) yields the hyperbolic Gauss map. This provides one (of at least two) possibilities to define discrete horospherical nets -- as analogs of smooth cmc-1 surfaces in hyperbolic space -- as special discrete isothermic nets: note that the hyperbolic Gauss map, being part of the definition, determines the hyperbolic geometry the surface is horospherical in as a subgeometry of Moebius geometry. The displayed model was obtained as a Darboux transform of a spherical discrete isothermic net with high symmetry. It therefore is a horospherical net, and can be considered as a discrete analog of a smooth surface of constant mean curvature 1 in hyperbolic space: in the picture, the sphere at infinity of hyperbolic space sits inside the surface (the surface having two ends) -- the standard Poincare ball model of hyperbolic space is obtained by inverting the configuration at the infinity sphere.
Model produced with: Mathematica
|Keywords||Darboux transformation; discrete horospherical net; constant mean curvature; isothermic surface; discrete isothermic net|
|MSC-2000 Classification||53A10 (37K25)|
Submitted: Sun Sep 10 07:10:24 CET 2000.
Accepted: Mon Nov 20 17:06:57 CET 2000.
Strasse des 17. Juni 136