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Math Help - surface is regular if curvature is non zero

  1. #1
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    Question surface is regular if curvature is non zero

    alpha(u) is a unit speed space curve, the chart (x,U) where

    x(u,v) = alpha(u) + v*(alpha'(u)), (u,v) in U

    i want to show that this surface is regular if the curvature k(u): = || alpha''(u)|| of the unit speed curve alpha(u) is non zero

    does anyone know how i would go about this?
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  2. #2
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    Quote Originally Posted by dopi View Post
    alpha(u) is a unit speed space curve, the chart (x,U) where

    x(u,v) = alpha(u) + v*(alpha'(u)), (u,v) in U

    i want to show that this surface is regular if the curvature k(u): = || alpha''(u)|| of the unit speed curve alpha(u) is non zero

    does anyone know how i would go about this?
    Is it really alpha' (and not alpha'') in the equation? And could you please precise what U is? Thank you.
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  3. #3
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    Quote Originally Posted by Laurent View Post
    Is it really alpha' (and not alpha'') in the equation? And could you please precise what U is? Thank you.

    alpha(u) is a unit speed space curve, and use this curve to construct a tangent developable surface with chart (x,U) where

    x(u,v) = alpha(u) + v*(alpha'(u)), (u,v) in U

    where U = {(u,v) in R^2 : -infinity < u < infinity, v>0}

    i want to show that this surface is regular if the curvature k(u): = || alpha''(u)|| of the unit speed curve alpha(u) is non zero


    so it is really alpha' in the equation. And i have re-written the question to specify what U is. Thank you.

    the solution i got was
    Xu (cross-product)Xv = v*alpha''(u)  (cross-product)t, where i used t as t = alpha(u)' from the frenet serret equations.

    i replaced  alpha''(u) with the k curvature, wasnt too sure how else to do it

    and so as v>o, k must be non zero for the chart to be regular.
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  4. #4
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    Quote Originally Posted by dopi View Post
    the solution i got was
    Xu (cross-product)Xv = v*alpha''(u)  (cross-product)t, where i used t as t = alpha(u)' from the frenet serret equations.

    i replaced  alpha''(u) with the k curvature, wasnt too sure how else to do it

    and so as v>o, k must be non zero for the chart to be regular.
    In the Serret-Frenet system, there are three vectors, (T,N,B). We have T=\alpha'(u), and \alpha''(u)=\kappa N. Therefore, X_u\times X_v=(T+v\kappa N)\times T=v\kappa N\times T=v\kappa B and \|X_u\times X_v\|=v\kappa\neq 0.
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