WOLFRAM|DEMONSTRATIONS PROJECT

Gauss Map and Curvature

​
surface
-
x
2
+
3
x
+
2
y
normal
radius
0.2
pt
÷
area 0. ÷ area 0.1411 = 0.
⟶ curvature 0. = max 1.789 × min 0
The Gauss map maps the unit normal of a surface (on the right) to the unit sphere (on the left). The area surrounding the point on the surface is thus mapped to an area on the unit sphere. As the radius of the loop approaches zero, the ratio of these areas approaches the Gaussian curvature of the surface at the point, which is also equal to the product of the principal curvatures (the maximum and minimum curvatures of the normal sections through the points).