[Math] Integrate the gaussian curvature

differential-geometryintegrationsurfaces

Let $T$ be a torus. We have a parameterization by $((c+a \cdot \cos(v))\cos(u),(c+a\cdot \cos(v),a\cdot \sin(v))$ for $u,v \in [0,2\pi)$. The first fundamental form is given by $E=(c+a\cdot \cos(v))^{2}, F=0, G=a^2$

and the second fundamental form is given by $e=-(c+a\cdot \cos(v))\cos(v),f=0,g=-a$. Since gaussian curvature is equal to determinante of the 2. fundamental form divided by 1. first fundamental, the gaussian curvature is equal to $\frac{\cos(v)\cdot a }{a^2(c+a\cdot \cos(v))}$

Now I want to integrate the gaussian curvature over T.

Do I integrate the gaussian curvature over T correctly: $\int_{[0,2\pi)^2}\frac{\cos(v)\cdot a }{a^2(c+a\cdot \cos(v))} du dv$?

This integral seems to me quite uncomputable.

edit: I computed the gaussian curvature (probably) correctly, I found it on another site (see page 2,

Best Answer

Let $K$ be the Gauss curvature. You would like to compute $\iint_{T} K \: dA$. Please remember that

\begin{equation*} dA = \sqrt{EG-F^2} du \: dv = a (c+a\cos v)\: du \:dv. \end{equation*}

Since you have already computed $K = \frac{a\cos v}{a^2(c+a\cos v)}$,

\begin{equation*} \iint_{T} K dA = \int_{0}^{2\pi} \int_{0}^{2\pi} \frac{a\cos v}{a^2(c+a\cos v)} \cdot a (c+a\cos v) \:du \: dv \end{equation*}

EDIT: Added some more detail about computing surface integrals.

Related Solutions

[Math] Calculation of Gaussian curvature from first fundamental form

While it's true that the computation of $K$ is much easier in orthogonal and especially in isothermal coordinates, finding such coordinates requires solving a PDE. I don't think it would be worth the effort in this case. Your coefficients $E,F,G$ are about as simple as one could have, the partials are immediately found and the determinants are not too bad. Everything is quite doable by hand.

With apologies for using a proprietary system (Maple), here is the general Brioschi formula:

with(LinearAlgebra):
E := 1+v^2; F := 2*u*v; G := 1+u^2;
A := Matrix([[-diff(E,v,v)/2+diff(F,u,v)-diff(G,u,u)/2, diff(E,u)/2, diff(F,u)-diff(E,v)/2], [diff(F,v)-diff(G,u)/2, E, F], [diff(G,v)/2, F, G]]);
B := Matrix([[0, diff(E,v)/2, diff(G,u)/2], [diff(E,v)/2, E, F], [diff(G,u)/2, F, G]]);
K := simplify((Determinant(A)-Determinant(B))/(E*G-F^2)^2);

Output: $$K=\frac{u^2+v^2}{(1+u^2+v^2-3u^2v^2)^2}$$

[Math] An “application” of Gaussian curvature

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.

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[Math] Calculation of Gaussian curvature from first fundamental form

[Math] An “application” of Gaussian curvature

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