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 trace is the mean
curvature.
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).
