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Math Help - Torsion

  1. #1
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    Torsion

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

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

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

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

    But I can't see where the \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 \alpha ''' comes in...

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

    Now 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 <br />
||\alpha||=1, t=\alpha', n=\alpha''/||\alpha''|| and eventually 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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