Describe all curves in \mathbb{R}^3 with constant curvature \kappa (s) = K > 0 and constant torsion \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,

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

Differentiating we get:

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

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

The solution to this ODE is

\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 \vec{a} and stuff?