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Math Help - Question about torsion

  1. #1
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    Question about torsion

    If \alpha has constant torsion c, \ c \neq 0 show that \alpha = \int_{s_0}^sT \ dt + K where K \in \mathbb{R}^3 and T represents the tangent vector from the Frenet frame.

    I was thinking of representing \alpha through it's Taylor series but that got me nowhere. So I would have \alpha(s) = \alpha(0) + sT_0 + \kappa_0\frac{s^2}{2}N_0+\kappa_0\tau_0\frac{s^3}{  6}B_0. But this doesn't seem to help.

    Know I figured that since we are dealing with constant torsion we can just express it through the formula yielding:

    c= \frac{\det(\alpha', \alpha'', \alpha''')}{\|\alpha' \times \alpha''\|^2}

    now rearranging and using the dot product with B I get \|\alpha' \times \alpha''\|c = {\det(\alpha', \alpha'', \alpha''')} B

    Now I don't know If I'm on the right track or if this is even correct.
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  2. #2
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    don't need to have constant torsion. For any curve \alpha, \frac{d\alpha}{ds}=T. Intergrate both sides we get \alpha(s) = \int_{s_0}^s T dt + \alpha(s_0)
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