6.2: Introduction to Gaussian Curvature

Defining Gaussian curvature

We may ask for which values of u and v the Gauss mapping is singular. This will happen whenever N_u(u,v) and N_v(u,v) are linearly dependent. That is, when

N_u(u ,v) × N _v(u,v) = 0.

Note that since N(u,v) · N(u,v) = 1, if we differentiate both sides with respect to u we obtain

2N_u( u,v)·N( u,v) = 0

N_u(u ,v) × N _v(u,v) = Κ( u,v)X_u(u,v) × X _v(u,v).

Note that the Gaussian curvature Κ(u,v) is positive if the frame given by N_u(u,v) followed by N_v(u,v) gives the same orientation to the tangent plane as the frame X_u(u,v) followed by X_v(u,v) , and Κ(u,v) is negative if these two orientations differ. The Gaussian curvature Κ(u,v) is zero precisely when the Gauss mapping is singular and the vectors N_u(u,v) and N_v(u,v) are linearly dependent.

At this point, we point out the analogy between the way we just defined Gaussian Curvature on a surface and the way we defined total curvature of a space curve in lab 4. Recall that T'(t) = κ(t)s'(t)P(t). The integral of |κ| along a space curve with respect to arclength is known as the total curvature, and it is equal to the length of the tangential indicatrix, which lies on the unit sphere. That is,

|κ|ds = |κ(t)|s'(t)dt = |T'(t)|dt

Similarly, the integral of Κ(u,v) over a surface with respect to area is known as the total Gaussian curvature, and it is equal to the area of the normal image, which lies on the unit sphere. That is,

ΚdA = Κ(u,v)||X_u(u,v) × X_v||dudv = ||N_u(u,v) × N_v(u,v)||dudv

Surface Colored by Gaussian Curvature

In this demonstration, the coloring of the surface indicates the sign of the Gaussian curvature in that region. Areas of positive curvature are colored red; areas of negative curvature are colored blue; and areas of zero curvature are colored white. The torus in this demo is divided into two regions, an inner part and an outer part, which correspond to regions of negative and positive Gaussian curvature respectively. The demo can be used to look at the sign of the Gaussian curvature for any surface. For example, a sphere which has constant positive curvature would be colored completely red, while a plane which has Gaussian curvature zero would be colored completely white.

The main purpose of this demonstration is to visualize Κ(u,v) as a function of u and v, defined over the domain of the original surface. A second window contains the surface Κ(u,v) which is color-coded in the same manner.