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Given a differentiable function $\kappa(s)$, $s \in I$, show that the parametrized plane curve having $k(s)=k$ as curvature is given by:

$\alpha(s)=(\int cos(\theta(s))ds+a,\int sin(\theta(s))ds+b)$

where

$\theta(s) = \int k(s)ds + \gamma$

And that the curve is determined up to translation of the vector (a,b) and a rotation of the angel $\gamma$


Solution:

$T'(s)=\kappa(-sin(\theta(s)),cos(\theta(s))=\kappa N$

Which is easy to see as $\alpha(s)$ is arc length parameterized.

How do I show that this curve is uniquely determined up to translation of the vector (a,b) and rotation of the angel $\gamma$?

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