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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?

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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 $\kappa\ge 0$.)

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  • $\begingroup$ By regular, does it also mean there is no corner on the curve? $\endgroup$
    – monk-E
    Commented May 13, 2019 at 18:27
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    $\begingroup$ Yes, absolutely. If you allow only piecewise-smooth curves, you don't have a Frenet frame at the corners. $\endgroup$
    – Ted Shifrin
    Commented May 13, 2019 at 18:34
  • $\begingroup$ By the way, you might be interested in my free differential geometry text. $\endgroup$
    – Ted Shifrin
    Commented May 13, 2019 at 18:39
  • $\begingroup$ Thank you for the help. I will go through the book. $\endgroup$
    – monk-E
    Commented May 13, 2019 at 21:24

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