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Math Help - Principal curvatures of a surface

  1. #1
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    Principal curvatures of a surface

    Find the equations of the principal curvatures of the surface:

    x=u, y=v, z=f(x,y)


    I can see a method to do this but it looks like it will be algebraically difficult if not impossible. Can you suggest a simpler way?

    What I have come up with is:


    I can calculate the surface metric:

    a_{\alpha \beta}=\left[\begin{array}{cc}1+f_u^2&f_uf_v\\f_uf_v&1+f_v^2\en  d{array}\right]

    And I can calculate the unit normal:

    \vec n=\frac{-f_u \vec e_1-f_v \vec e_2+\vec e_3}{(f_u^2+f_v^2+1)^{\frac 12}}

    Next I would calculate the curvature tensor:

    b_{\alpha \beta}=-\frac{\partial \vec r}{\partial u^{\alpha}}\cdot\frac{\partial \vec n}{\partial u^{\beta}}


    But the partial derivatives of n are going to get ugly and I still would not be finished. I need the eigenvalues of:

    a^{\alpha \gamma}b_{\gamma \beta}
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  2. #2
    Super Member Rebesques's Avatar
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    It's going to be lengthy anyhow, but here's a slight improvement:
    Compute the Gaussian and mean curvatures K and H in terms of the coefficients in the first and second fundamental forms, and remember that
    the principal curvatures satisfy the equation x^2-2Hx+K=0.
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