TOPICS
Search

Gauss Map


The Gauss map is a function N from an oriented surface M in Euclidean space R^3 to the unit sphere in R^3. It associates to every point on the surface its oriented unit normal vector. Since the tangent space at a point p on M is parallel to the tangent space at its image point on the sphere, the differential dN can be considered as a map of the tangent space at p into itself. The determinant of this map is the Gaussian curvature, and negative one-half of the trace is the mean curvature.

GaussMap
GaussMapReIm
GaussMapContours

Another meaning of the Gauss map is the function

 f(z)=1/z-|_1/z_|

(Trott 2004, p. 44), where |_z_| is the floor function, plotted above on the real line and in the complex plane.

GaussMap2
GaussMap2ReIm
GaussMap2Contours

The related function frac(1/z) is plotted above, where frac(z) is the fractional part.

GaussMapAbs

The plots above show blowups of the absolute values of these functions (a version of the left figure appears in Trott 2004, p. 44).


See also

Curvature, Fractional Part, Gaussian Curvature, Mean Curvature, Nirenberg's Conjecture, Patch

Portions of this entry contributed by John Derwent

Explore with Wolfram|Alpha

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/.

Referenced on Wolfram|Alpha

Gauss Map

Cite this as:

Derwent, John and Weisstein, Eric W. "Gauss Map." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GaussMap.html

Subject classifications