Gauss Map

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).

REFERENCES:

Gray, A. "The Local Gauss Map" and "The Gauss Map via Mathematica." §12.3 and §17.4 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 279-280 and 403-408, 1997.

Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, 2004. http://www.mathematicaguidebooks.org/.