What is Gaussian curvature?

What is Gaussian curvature?

The definition of Gaussian curvature in Wikipedia is as follows

In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ₁ and κ₂, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way it is isometrically embedded in space. This result is the content of Gauss's Theorema egregium.

Symbolically, the Gaussian curvature Κ is defined as

K= k1 x k2.

where κ₁ and κ₂ are the principal curvatures.

I will explain the above definition with a simple example. We will consider a cylindrical surface and try to find the Gaussian curvature of that surface.

By the definition K=k1 x k2

We need k1 and k2 of each point where k1 and k2 are the min and max values respectively.

We create a point on the surface first, then a normal to the surface passing through the point and a tangent plane to the surface passing through the point.

See the image below.

We need to have few planes (say 8 here) to create Normal section curves. Means, we are going to create intersection curves using planes which pass through the surface normal and Normal to the Tangent plane. Our point on the surface is common to all Normal section curves. All the Normal section curves contain our point on surface.

See the figure below. First figure shows the planes for creating intersection curves and the second one shows the Normal section curves created using the Planes.

DTM1, DTM2, DTM3, DTM4 etc are the planes.

Next figure you can see the Normal Section curves.

Make a note that the curve highlighted in red is a straight line whose curvature is zero, means the k2 is zero (in this case).

Next I would like to see the maximum curvature (k1) at the point. I have curvature plot tool and right of the bat I will highlight the curvature combs of the above curves.

Now I have the curvature variation of all the 6 curves, see below

How will I find the maximum curvature at the point on surface? I know the parameter value for our point is 0.5 (point is at the mid of the curves you can see)

Curve Parameter	Curvature
0.5	0.006666667
0.5	0.002988536
0.5	0.002526927
0.5	0.00063661

I found that the maximum curvature is 0.0067. The value of k1, if we consider only the above six curves will be 0.0067(Correspond to the circle)

But when the software calculates the maximum curvature, it looks for all the possible curves that can pass through our point on the surface.

Now what is K?

K= k1 x k2

= 0.0067 x 0

= 0

You take any point on the cylindrical surface the value of k2 will be zero corresponding to the straight line.

So K=0 over the entire cylindrical surface.

Now see the Gaussian Curvature display with color contour

The green color has a value of 0, you can see from the color spectrum.

A positive Gaussian curvature value means the surface is locally either a peak or a valley. A negative value means the surface locally has a saddle points. And a zero value means the surface is flat in at least one direction.

For a flat surface the Gaussian curvature will be zero. How? Self explanatory.

Thanks

PADMAKUMAR NAIR