In a question, it's desired to find the curvature of a curve
$γ_r(t)=γ(t)+rN(t)$
in terms of $κ$ (the curvature of the planar curve $γ$) and the real number $r$. ($N(t)$ is the normal vector of $γ$).
Does it really have a good representation? I tried so much but didn't succeed!
It reminded me the concept of Bertrand curve. it didn't help me, though.
I think it's also OK if it includes tangent vecotor, normal vector, derivatives of the curve, etc. But probably the simplest answer is considered.
On
In simpler terms for plane curves $ (\tau =0) $ the Frenet- Serret relations ( primed on arc )
$$ N' = -\kappa T; T'= \kappa N ;\to N''=- \kappa^2 N $$
$$\kappa^2= -\frac{N''}{N} \text{ wrt arc}$$
Wrt general parameterization $t$ see at bottom of wiki Curvature
For rectangular coordinates when normal makes $\phi$ to positive x-axis
$$ \phi= \tan^{-1}\frac{dy}{dx}\;; \kappa = \phi'.$$
First show that $$\dot\gamma_r(t)=\dot\gamma(t)\bigl(1-r\kappa(t)\bigr).$$ Compute $\ddot\gamma(t)$. You'll arrive in a pretty nice expression for $\kappa_r$ which includes only $\kappa$ and $r$, but not the derivative of $\kappa$.
Hint: use $$\kappa(t)=\frac{\langle\ddot\gamma,J\dot\gamma\rangle}{\|\dot\gamma\|^3},$$ where $J(a,b):=(-b,a)$.