Computation of the curvatures of a torus as surface of revolution
A Torus and its Curvatures
Let's consider a torus, parametrized as a surface of revolution and compute its different curvatures.
First Step, all the partial derivatives we'll need.
For all of our computations, we'll want the and partial derivatives of of order 1 and 2. So we compute them and store them in some variables.
First Fundamental Form
The Normal Vector
The Second Fundamental Form
Now we have enough data to compute the second fundamental form. We don't have script letters here, so I'm gonna cheat.
The Shape Operator
Finally, curvatures
The Gaussian curvature is the determinant of the shape operator:
The mean curvature is half the trace of the shape operator:
...and the principal curvatures are the eigenvaules of the shape operator. In this case, the matrix is diagonal, so these are easy to read off. But generally, one could use this command:
and if for some reason that command won't work for you, you can always do it by hand: