# Frenet Equations and Evolutes

Let $c=r+(1/\varkappa)N$ be the evolute of a parametric equation $r$ with curvature $\varkappa$ and normal unit vector $N$. Show that this is true if $(r-c)\cdot T=0$, and $(r-c)\cdot N=-1/\varkappa$, where $T$ is its unit tangent vector.

To show this, what I did was notice the following:

\begin{align} &(r-c)\cdot N=-1/\varkappa,\\ \Longrightarrow&(r-c)\cdot N=(-1/\varkappa)N\cdot N,\\ \Longrightarrow&r-c=(-1/\varkappa)N,\\ \Longrightarrow&c=r+(1/\varkappa)N. \end{align}

Firstly, I do not know if this is valid; and secondly, I do not know where to use the fact that $(r-c)\cdot T=0$.