6.3: Investigating Gaussian Curvature

First consider the Gaussian curvature of a function graph X(u,v) =(u,v,f(u,v)) defined over a rectangular domain in the plane. We can observe a qualitative difference between positive and negative Gaussian curvature of a surface by looking at the images in the Gauss map of the four corners of a rectangular region in counterclockwise order. When the Gaussian curvature is positive, the image of the four corners of the region under the normal mapping of the surface will also be traced in a counterclockwise manner. However, if the Gaussian curvature is negative, the image of the boundary of the rectangle will be traced in a clockwise manner. When the Gaussian curvature is zero somewhere in the rectangular domain, the normal mapping behaves in a more complicated way.

Gaussian Curvature of Quadratic Function Graphs

Besides the domain and the quadratic surface, this demo displays the Normal Map of the Surface, and the Gaussian Curvature Graph over the domain. The variables a,b, and c in the control panel allow you to change the coefficients of a general quadratic function of two variables. The two white dots (or hotspots) in the domain window can be used to change either the size or location of the rectangle.

The rotation of the surface and the normal map in this demo is linked, which means that the normal map rotates identically as you rotate the surface. This ensures that you are looking at the image of the rectangle from the same viewpoint in both the windows, so you can properly observe the relative orientation of the rectangle's corners.

The demos in this lab show a square in the domain that is divided into four quadrants colored red, magenta, blue and white. The colors of its image on the surface and on the normal map are marked the same way to help visualize the difference between positive and negative Gaussian curvature.

Exercises

1. Analyze the Gaussian curvature of the function graph where f(u,v)=au²+buv+cv² . For which values of a, b, and c will the curvature be positive and for which will it be negative?

2. If a = c = 1, for which values of b will the Gaussian curvature be zero?

The surfaces you will be asked to investigate are all predefined in the library of standard surfaces. They should all be used with the standard u- and v- domains, which run from -1 to 1.

This next demo generalizes the previous one to allow you to input any surface.

Exercises

(Investigate the Gaussian curvature of the following families of surfaces as the value of c changes)

1. The Shoe surface, where f(u,v)=1 3u³-c2v²

2. The Biparabola surface, where f(u,v)=cu⁴+u ²v-v²

3. The PerturbedMonkeySaddle, where f(u,v)=1 3u³-uv²+c(u ²+v²)
(Where and when does the Gauss map of this surface have singularities?)

4. The Folded Handkerchief surface, where f(u,v)=1 3u³+uv²+c(u ²-v²)
(Observe the normal map at small values of c to see where it gets its name.)

Gaussian Curvature on Surfaces of Revolution

Now consider the Gaussian curvature for a general surface of revolution. A general surface of revolution is given by the following formula:

X(u,v) = (r(v)cos(u),r(v)sin(u),h(v))

This parametrization gives the surface of revolution obtained from revolving the curve X(t) = (r(t),h(t)) in the xz-plane (or yz-plane) around the z-axis. For example, a torus may be obtained with (r(t),h(t)) = (a+bcos(t),bsin(t)), where a and b are constants with 0<b<a.

This next demo is like the previous two, but allows you to input r(v) and h(v) to create a surface of revolution.

Exercises

1. The Bell surface, where r(v)=v and h(v)=cos(v), with 0≤u≤2p.

2. The Torus of revolution, where r(v)=a+bcos(v) and h(v)=bsin(v), with 0<b<a and 0≤u,v≤2p.