We have that $$\epsilon (s)=\gamma (s)+\frac{1}{\kappa\_s(s)}\cdot n\_s(s)$$ 

We regard $\epsilon$ as the parametrization of a new curve, called the evolute of $\gamma$ (if $\gamma$ is any regular plane curve, its evolute is defined to be that of a unit-speed
reparametrization of $\gamma$). 

Assume that $\dot \kappa\_s (s) \neq 0$ for all values of $s$ (a dot denoting $\frac{d}{ds}$ ), say $\dot \kappa\_s >0$ for all $s$ (this can be achieved by replacing $s$ by $−s$ if necessary). 

Show that the arc-length of $\epsilon$ is $\frac{-1}{\kappa\_s(s)}$ (up to adding a constant), and calculate the signed curvature of $\epsilon$. 

Show also that all the normal lines to $\gamma$ are tangent to $\epsilon$ (for
this reason, the evolute of $\gamma$ is sometimes described as the ‘envelope’
of the normal lines to $\gamma$). 

Show that in particular case of the cycloid ( $a > 0$ is a constant),

$$\gamma (t) = a(t − \sin t, 1 − \cos t), 0 <t< 2\pi ,$$ 

the evolute is

$$\epsilon (t) = a(t + \sin t, −1 + \cos t), $$ 

and that after a suitable re-parametrization $\epsilon$ can be obtained from $\gamma$ by a translation of the plane.

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I have shown that the arc-length of $\epsilon$ is $\frac{-1}{\kappa\_s(s)}$ up to adding a constant. 

Could you give me a hint how we could calculate the signed curvature of $\epsilon$ ? 

Also how could we show that all the normal lines to $\gamma$ are tangent to $\epsilon$ ?