Isothermic cmc-1 Cylinder

Author

Description

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

References

1. A. Bobenko, U. Pinkall: Discrete isothermic surfaces, J. reine angew. Math. 475 (1996), 187-208.
2. 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).
3. 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.
4. U. Hertrich-Jeromin, E. Musso, L. Nicolodi: Moebius geometry of surfaces of constant mean curvature 1 in hyperbolic space (1998), preprint.
5. U. Hertrich-Jeromin: Transformations of discrete isothermic nets and discrete cmc-1 surfaces in hyperbolic space (1999), preprint.

Files

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

Submission information

Submitted: Sun Sep 10 07:10:24 CET 2000.
Accepted: Mon Nov 20 17:06:57 CET 2000.

Author's Address

TU Berlin
Strasse des 17. Juni 136
D-10623 Berlin
Germany
udo@sfb288.math.tu-berlin.de
http://www-sfb288.math.tu-berlin.de/~udo/