Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

This is not a homework question.

Just wondering if there is a general formula for the gaussian curvature at point (x,y,f(x,y)) in terms of x, y, and f(x,y).

I didn't see any thing like that on the wikipedia article, but maybe it's hidden behind all these letters/symbols which are not actually described.

By the way, I'm studying computer science. For us, a general formula is one that works most of the time. When it doesn't work, we call this an exception, and it requires special handling :). So here of course, the function is assumed everywhere indefinitely continuous and differentiable and bounded inside any box that would fit the RAM - whatever that means - and also in R3, inside a standard classical euclidean space...

Thank you.

share|improve this question
add comment

2 Answers

up vote 4 down vote accepted

There are a few explicit formulas at http://mathworld.wolfram.com/GaussianCurvature.html.

share|improve this answer
Right. When believing it is there, it is easier to find it. Thanks. –  Gzorg Oct 22 '09 at 21:34
add comment

If the surface is in Monge form, i.e. written as the graph of a function $z = f(x,y)$ such that $f(0,0) = f_x(0,0) = f_y(0,0) = 0,$ then there is an easy way to calculate the Gauss curvature. You take a rotation about the $z$-axis which kills off the $xy$-term in the quadratic part of $f.$ Then you will be able to write the surface as $z = \frac{1}{2}(\kappa_1x^2 + \kappa_2y^2) + \cdots,$ where the dots denote terms of order three and above. A rotation is a special orthogonal transformation and does not change the Gauss curvature.

The $\kappa_1$ and $\kappa_2$ will be in terms to the coefficients in the Taylor series of the original expression for $f.$

The numbers $\kappa_1$ and $\kappa_2$ are the principal curvatures of the surface at $(0,0,0).$ Moreover, the $x$-axis and the $y$-axis are the principal directions of the surface at $(0,0,0).$ (Recall that there are always two orthogonal principal directions at each non-umbilic point, i.e. $\kappa_1 \neq \kappa_2,$ of the surface.)The Gauss curvature is given by $K = \kappa_1\kappa_2$ and the mean curvature is given by $H = \frac{1}{2}(\kappa_1 + \kappa_2).$

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.