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Thread: Constant curvature and constant torsion

  1. #1
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    Constant curvature and constant torsion

    Describe all curves in $\displaystyle \mathbb{R}^3$ with constant curvature $\displaystyle \kappa (s) = K > 0$ and constant torsion $\displaystyle \tau (s) = T$.
    I know the solution is a helix, but it's proving it that's the hard part.

    So the first thing I did was use the Frenet Equations,

    $\displaystyle \vec{n}'(t) = - \kappa \vec{t} + \tau \vec{b}$

    Differentiating we get:

    $\displaystyle \vec{n}''(t) = - \kappa^2 \vec{n} - \tau^2 \vec{n}$

    $\displaystyle \vec{n}''(t) = - \underbrace{[\kappa^2 + \tau^2]}_{\omega ^2} \vec{n}$

    The solution to this ODE is

    $\displaystyle \vec{n}(t) = \vec{a} \sin (\omega t) + \vec{b} \cos (\omega t)$

    Now I can integrate twice to get the parametrisation, but how do I solve for the constant vectors $\displaystyle \vec{a}$ and stuff?
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  2. #2
    Super Member Rebesques's Avatar
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    Re: Constant curvature and constant torsion

    you don't have to compute anything about the constant vectors; any value they assume will describe such a curve, and vice versa.
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