I'm trying to help a collegue in his differential geometry exam. His exam assignment is to discuss gaussian curvature and some of its properties and applications.
I know that the gaussian curvature is defined as the product of the two principal curvatures. I know that it can be computed as
$$K(p)=\frac{eg-f^2}{EG-F^2},$$
with the terms in the numerator being the components of the metric of the second fundamental form, the denominator terms being ditto in the first fundamental form. I know that $K(p)>0$ at an umbilical point, $K(p)<0$ at a saddle point, $K(p)=0$ at a point that's "flat" in one or both directions. I had overlooked the Frenet-Serret theorem about triangles on surfaces, but I found that too, now.
What I don't know is just what constitutes an "application".
I might be overthinking this. What do you reckon?
Best Answer
Gaussian curvature has several applications in engineering and manufacturing. Specifically, a surface whose Gaussian curvature is zero is developable, which means that it can be formed from a flat sheet without any stretching or tearing. Planes, cylinders and cones are the most obvious developable surfaces, but there are others, too. All developable surfaces are ruled surfaces, but the converse is not true.
Why does this matter? Here are a few examples:
Suppose I'm making the hull of a ship from flat metal plates. It's not too difficult to bend these plates, but it's very difficult to stretch them, so developable surfaces are better for ship hulls.
Suppose I'm designing a bottle that will have a paper label glued to its outside. The label can be bent, but not stretched, so the area of the bottle where it is afixed needs to be developable.
Suppose I'm going to manufacture a ruled surface by "swarf cutting". This means that the shape is formed by cutting with the side of a cylindrical cutting tool. At each step, the tool will contact the surface along a straight line "ruling", and the surface normal will be the same at all points along this line. It's not obvious, but it turns out that this implies that the surface is developable. So, if you want to manufacture a ruled surface by swarf cutting (which is desirable, because it's cheap and fast), then you should make it developable. There's a nice picture of swarf cutting on this page, in the section entitled "Advanced 5D Milling".
More generally, if Gaussian curvature is not zero, its magnitude gives you some information about "how far from developable" the surface is. This in turn tells you how much stretching will be needed to form it. If the stretching is too much, the material will tear. This is often an issue when stamping sheet metal to form car bodies, for example. Sometimes the body panel has to be redesigned to avoid the tearing.
Regarding what constitutes an "application" ... when mathematicians say "application", they often mean using some theorem to prove other results, in other areas of mathematics. This always seems a bit incestuous, to me; I much prefer "applications" in the real world.