# Finding a plane curve with curvature $\kappa(s)$

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 $$\gamma$$?