The Gauss map is a function 
from an oriented surface 
in Euclidean
space 
to the unit sphere in 
. It associates to every point
on the surface its oriented unit
normal vector. Since the tangent space at a
point 
on 
is parallel to the tangent space at its image point on the sphere, the differential
can be considered as a map
of the tangent space at 
into itself. The determinant
of this map is the Gaussian
curvature, and negative one-half of the matrix trace
is the mean curvature.
For an oriented surface in 3-dimensional hyperbolic space, the two directions of the unit normal vector determine geodesics ending at the ideal boundary. The resulting endpoint maps are the hyperbolic Gauss maps (Epstein 1986).
Another meaning of the Gauss map is the function
 |
(Trott 2004, p. 44), where 
is the floor function,
plotted above on the real line and in the complex
plane.
The related function 
is plotted above, where 
is the fractional part.
The plots above show blowups of the absolute values of these functions (a version of the left figure appears in Trott 2004, p. 44).
