in 2D dimensional plane, is it problematic to have Frenet-Serret frame with zero curvature?

Question

I have a Frenet-Serret frame moving on a 2-D plane. As of now, I do not care about the binormal vector. So my equations are given by,

\begin{align} \dot{T} = v\kappa N \\ \dot{N} = -v\kappa T \end{align}

Here $v$ is the constant speed and $\kappa$ is the curvature. I don't see any problem with these equations if $\kappa = 0$, but I have read that the frame is not defined if curvature is zero.

Can anyone please explain it?

Answer 1

As long as the curve is regular (has nonzero velocity at each point) you can always define a right-handed $T,N$ frame everywhere. But you must allow curvature to change sign. (Ordinarily, we always define $\kappa\ge 0$.)