Computing principal curvatures in local coordinates

Question

I was wondering how to compute the principal curvatures when working with a local parametrization. (Context: Surfaces, i.e. 2-dim Manifolds). Since I have formulas for the Gaussian and the mean curvature, I used them to derive:

$$k_i = H \pm \sqrt{H^2 - K} \, ,$$

where the $k_i$ denote the two principal curvatures, K the Gaussian curvature and H the mean curvature. Now when looking through the Wikipedia page about Differential geometry of surfaces I saw a different formula (in the bottom row of the table I have linked):

$$k_i = \frac{H \pm \sqrt{H^2 - 4K}}{2}.$$

Now I have two questions:

1. Why is the formula different? Is it a different context? As far as I could tell everything else matches what I am working with...

2. Is there a better way to compute the principal curvatures when working with local coordinates? Using the formulas for the Gaussian and mean curvature is pretty straight forward, which I like, but it does involve a lot of computation. Maybe there is a more effective way to do it?

Thanks!

Answer 1

1. It seems that in your formulas the definition of the mean curvature differs. There is a choice to be made and both options are equally valid and are used in the literature. One way would be be $$\frac{\kappa_1+\kappa_2}{2}=H,$$ i.e. $H$ is the mean of the principal curvatures. The other option is that $H$ is the sum of the principal curvatures, i.e. $$\kappa_1+\kappa_2=H.$$ Wikipedia uses this version. Since the underlying math and geometry is not really changed, both version are valid. Only the calculations may differ here and there.

2. You could calculate the principal curvatures as the eigenvalues of the second fundamental form. It may be a bit shorter, see https://en.wikipedia.org/wiki/Principal_curvature

Answer 2

It is just a matter of convention on what you call the mean curvature $H$.

Suppose $\Sigma^2\subset \mathbb{R}^3$ is an embedded surface and $S$ is its shape operator. Then the principal curvatures are the eigenvalues of $S$ (and the principal curvature directions are the corresponding eigen directions).

Some author refer to the mean curvature $H$ as the trace of $S$, that is, the sum of the principal curvatures, while some others refer to $H$ as the mean of the principal curvatures, that is, in dimension $2$, $\frac{1}{2}\mathrm{trace}S$.

Changing $H$ by $H/2$ in your first formula gives the second.

Concerning the computations of the principal curvatures: it is in general a hard and impossible problem because it leads to computing eigenvalues of an endomorphism and roots of a polynomial, which we know is impossible in degree greater than 5. But in dimension $2$, the formula you gave is just the usual formula for computing the roots of a degree 2 polynomial!