I am trying to find the principal curvature and its direction of:

$$x^2+y^2-z^2=0$$ I tried to re-parametrize with $r(\theta,z)=(z^2\cos(\theta),z^2\sin(\theta),z^2)$

But the second fundamental matrix get a full zero 2nd column:

The normal being =$N(\theta,z)= \frac{1}{\sqrt{2}}\,\,\,\,\,\matrix{\cos(\theta)\\ \sin(\theta)\\-1}$

and 2nd fundamental form being: $\matrix{z\,\frac{\sqrt{2}}{2}&0\\0&0}$

Am I using the wrong parametrization or I am totally off point? As I want to compute the determinant of the operator $g^{-1}*B$ where $g$ is the first fundamental and $ B$ is the 2nd to get the principal curvature. Any help is appreciated.


1 Answer 1


The surface you are evaluating is an infinite cone which will have a curvature in one principal direction that is a function of the current 'radius' of the cone (i.e. z) and zero curvature in the other principal direction. The fact that your second fundamental form only has diagonal entries, which are the eigenvalues, is consistent with this.


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