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Thread: Torsion

  1. #1
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    Torsion

    I'm looking to derive the equation for torsion for a space curve $\displaystyle \alpha (u)$

    $\displaystyle \tau= \frac{(\alpha ' \times \alpha '')\cdot \alpha '''}{{||\alpha '\times \alpha ''||}^{2}}$

    I think I use $\displaystyle b = t \times n $ and of course $\displaystyle t = \frac{\alpha '}{||\alpha '||}$

    Then $\displaystyle \frac{db}{ds}=\frac{1}{||\alpha '||}\cdot\frac{db}{du}=\tau n$

    But I can't see where the $\displaystyle \alpha '''$ comes in...
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  2. #2
    Super Member Rebesques's Avatar
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    But I can't see where the $\displaystyle \alpha '''$ comes in...

    From $\displaystyle \frac{db}{ds}$.
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  3. #3
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    First I need $\displaystyle \frac{db}{du}=\frac{d}{du}(t \times n) = t' \times n + t \times n'$

    Now $\displaystyle n = \frac{t'}{||t'||}$.

    How do I find t' and n' ?
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  4. #4
    Super Member Rebesques's Avatar
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    Well... First of all, you can reparametrize for arc length, so that $\displaystyle
    ||\alpha||=1$, $\displaystyle t=\alpha'$, $\displaystyle n=\alpha''/||\alpha''||$ and eventually $\displaystyle b'=\left(\alpha'\times\frac{\alpha''}{||\alpha''|| }\right)'$, etc...
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  5. #5
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    What if the aim is to find the torsion for any parametrisation?
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