Finding a curve from its evolute

Question

Consider the evolute given by $$ \gamma: I \subset \mathbb{R} \to \mathbb{E}^2: t \mapsto (\cos(t),\sin(t))$$ Now, how do I find all the curves $\alpha$ that have $\gamma$ as their involute? I tried using the fact that an involute is given by $$\gamma(t) = \alpha(t) + \frac{1}{\kappa(t)}N(t)=(\cos(t),\sin(t)) $$ so that also $$\gamma'(t) = \left(\frac{1}{\kappa(t)}\right)'N(t)= (-\sin(t),\cos(t))$$ and then using the Frenet-Serret formula's, but I haven't gotten very far...

Answer 1

Hint: The involute of an evolute is the original curve. and the evolute of an involute is the original curve.