The hyperbolic Gauss maps of an oriented immersion into 3-dimensional hyperbolic space assign to each point the two ideal points approached by the geodesic through that point in the directions of the positive and negative unit normal vector. Thus they are maps
 |
Reversing the orientation interchanges 
and 
. When only one orientation is under consideration, the corresponding
endpoint map is often called the hyperbolic Gauss map.
Unlike the ordinary Gauss map of a surface in 3-dimensional Euclidean space, a hyperbolic Gauss map takes values on the ideal boundary rather than in a sphere of normal vectors. Hyperbolic Gauss maps play an important role in the geometry of horospheres, surfaces of constant mean curvature, and hyperbolic minimal surfaces (Epstein 1986, Bryant 1987).
