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) | |

Zentralblatt No.
| 01682977 |

- A. Bobenko, U. Pinkall:
*Discrete isothermic surfaces*, J. reine angew. Math.**475**(1996), 187-208. - U. Hertrich-Jeromin, T. Hoffmann, U. Pinkall:
*A discrete version of the Darboux transform for isothermic surfaces*, in A. Bobenko, R. Seiler (Eds.): Discrete integrable Geometry and Physics, Oxford UP (1999). - U. Hertrich-Jeromin:
*The surfaces capable of division into infinitesimal squares by their curves of curvature: A nonstandard-analysis approach to classical differential geometry*, Math. Intell.**22**(2000), 54-61. - U. Hertrich-Jeromin, E. Musso, L. Nicolodi:
*Moebius geometry of surfaces of constant mean curvature 1 in hyperbolic space*(1998), preprint. - U. Hertrich-Jeromin:
*Transformations of discrete isothermic nets and discrete cmc-1 surfaces in hyperbolic space*(1999), preprint.

- Master File: DarbouxSphere_Master.jvx
- Applet File: DarbouxSphere_Applet.jvx
- Preview: DarbouxSphere_Preview.gif

Submitted: Sun Sep 10 07:10:24 CET 2000.

Accepted: Mon Nov 20 17:06:57 CET 2000.

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