I have the parametrization
$x(u,v)=(\cos u \sin v, \sin u \sin v , \cos v+\log (\tan {v/2}))$
with $0<v<\pi $ , $0<u<2\pi$.
From this parametrization, how can I compute (optimally) the Gaussian curvature?
I know for example that the pseudosphere is a revolution surface, then there should exist a more easy way to calculate its curvature.
Thanks!
Best Answer
Take $\partial_1=\frac{\partial x}{\partial u}$ and $\partial_2=\frac{\partial x}{\partial v}$ as the tangent frame. Then the normal will be $N=\frac{\partial_1\times\partial_2}{||\partial_1\times\partial_2||}$. Now the derivatives $D_{\partial_1}N=\frac{\partial N}{\partial u}$ and $D_{\partial_2}N=\frac{\partial N}{\partial v}$ are going to be tangent too, so you will get a base change $$D_{\partial_1}N=A\partial_1+B\partial_2$$ $$D_{\partial_2}N=C\partial_1+D\partial_2$$ for some scalars $A,B,C,D$. Then the determinant $AD-BC$ is the Gaussian curvature.