I have started to study differential geometry and have some questions about an exercise which is probably not very difficult.
Exercise: Let $\gamma: I \rightarrow\mathbb{R}^{2}$ be a regular curve, parametrized by arclength, with Frenet frame $\{ T(s), N(s) \}$. For $\lambda \in \mathbb{R}$ we define the parallel curve $\gamma_{\lambda}: I \rightarrow \mathbb{R}^{2}$ by
$\gamma_{\lambda}(t) = \gamma(t) + \lambda N(t)$.
Calculate the curvature $\kappa_{\lambda}$ of those curves $\gamma_{\lambda}$ which are regular.
Attempt to solution:
First, we need to now when $\gamma_{\lambda}$ is regular, i.e., when $\dot{\gamma}_{\lambda} \neq 0$.
$\dot{\gamma}_{\lambda} = \dot{\gamma}(t) + \lambda \dot{N}(t)$,
so if $\dot{\gamma}(t) \neq -\lambda\dot{N}(t)$,
then $\gamma_{\lambda}$ is regular.
Secondly, we need to calculate the curvature, and we have that $\kappa(t) = \langle\dot{T}(s), N(s)\rangle$, if $\{ T(s), N(s) \}$ is the Frenet frame for a curve $\gamma(s)$, if $\gamma$ is parametrized by arclength.
However, our curve $\gamma_{\lambda}$ is not parametrized by arclength so I don't think we can use the scalar product directly to calculate the curvature. So I am not sure how to continue.
I guess we shall do something like $\langle \dot{T}_{\lambda}, N_{\lambda}\rangle$.
Any help would be appreciated!