> 
> Given a differentiable function $k(s)$, $s\in I$, show that the parametrized plane curve having $k(s)=k$ as curvature is given by
>  $$ \alpha (s) = \left( \int \cos\theta(s)ds + a, \int \sin\theta(s)ds + b \right) $$
>  where $$ \theta(s)= \int k(s)ds + \varphi $$ and that the curve is determined up to a translation of the vector $(a,b)$ and a rotation of the angle $\varphi$.
> 
> 
> 

This exercise is from Do Carmo *Differential Geometry of Curves and Surfaces*, section 1.5.
My problem is that I'm not even sure where to start. It is not clear for me what is required for this type of proofs. I mean, the proof is based on some constructive procedure? Or the usual way is, instead, start as "if such $\alpha$ exist, it must verify such and such..."