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Math Help - Find curvatures

  1. #1
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    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.
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  2. #2
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    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.
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  3. #3
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    Addition

    A moment of additional thought made me realize that the parametrization is simpler with hyperbolic functions: the upper sheet can be done as:
    (x,y,z)=(\sinh{u}\cos{v},\sinh{u}\sin{v},\cosh{u})
    with u\ge0,\,0\le{v}<2\pi as before.

    --Kevin C.
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