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Thread: Relation between curvature and torsion

  1. #1
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    Relation between curvature and torsion

    In the context of tensor calculus, by using Serret-Frenet formula or otherwise,
    how to prove that


    $\tau^2=\displaystyle\frac{r'''^2}{k^2}-k^2-(\frac{k'}{k})^2$
    where $\tau$ and $k$ represent respectively torsion and curvature.
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  2. #2
    Super Member Rebesques's Avatar
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    Re: Relation between curvature and torsion

    You may consider the curve parametrized for arc length.
    The formulas for curvature and torsion become

    k(t)=||r''(t)||, \ \tau(t)=(r'(t),r''(t),r'''(t))/k^2(t)


    You can substitute these in the right hand of the equation and obtain the left hand side.
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