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Given differentiable functions $k(s),\tau(s)$ with domain $s\in (a,b)$, there exist a differentiable function $\gamma:(a,b)\to \mathbb R^3$ such that $k,\tau$ are it's respectively curvature and torsión.

My question: I need an example of a pair $k,\tau$ such that the curve it's closed, but if we consider a new curve $\gamma_1$ that has the same curvature $k$, but torsión $\tau=0$, $\gamma_1$ it's not closed.

Eustass
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    Just about any curve with torsion will do the trick, so find something that loops around a bit in a funny way, compute its curvature and torsion, then look at just the curvature and see how the new curve looks. I say just about any curve because the probability a random curve with torsion will meet up again when you remove its torsion component is effectively $0$. – muzzlator Mar 26 '13 at 03:53

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