# Find curvatures

• April 19th 2008, 08:34 PM
Find curvatures
Find the gaussian and the mean curvatures of $X^2 + y^2 = Z^2 - 1$

Now, I know that I have to parametrize this equation first, but I'm not sure how I should do it, please help.
• April 26th 2008, 01:47 PM
TwistedOne151
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
$(x,y,z)=(u\cos{v},u\sin{v},\sqrt{1+u^2})$
with parameters $u\ge0,\,0\le{v}<2\pi$ (note that u and v correspond to r and θ of cylindrical coordinates).

--Kevin C.
• April 26th 2008, 02:01 PM
TwistedOne151
$(x,y,z)=(\sinh{u}\cos{v},\sinh{u}\sin{v},\cosh{u})$
with $u\ge0,\,0\le{v}<2\pi$ as before.