Find the gaussian and the mean curvatures of $\displaystyle X^2 + y^2 = Z^2 - 1 $

Now, I know that I have to parametrize this equation first, but I'm not sure how I should do it, please help.

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- Apr 19th 2008, 08:34 PMtttcomraderFind curvatures
Find the gaussian and the mean curvatures of $\displaystyle X^2 + y^2 = Z^2 - 1 $

Now, I know that I have to parametrize this equation first, but I'm not sure how I should do it, please help. - Apr 26th 2008, 01:47 PMTwistedOne151
Isn't this a hyperboloid of two sheets? Are we interested in only one of the sheets, or both? As for parametrizing, consider cylindrical coordinates: that is, we can parametrize the upper sheet as

$\displaystyle (x,y,z)=(u\cos{v},u\sin{v},\sqrt{1+u^2})$

with parameters $\displaystyle u\ge0,\,0\le{v}<2\pi$ (note that u and v correspond to r and θ of cylindrical coordinates).

--Kevin C. - Apr 26th 2008, 02:01 PMTwistedOne151Addition
A moment of additional thought made me realize that the parametrization is simpler with hyperbolic functions: the upper sheet can be done as:

$\displaystyle (x,y,z)=(\sinh{u}\cos{v},\sinh{u}\sin{v},\cosh{u})$

with $\displaystyle u\ge0,\,0\le{v}<2\pi$ as before.

--Kevin C.