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?


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 May 13 '19 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 May 13 '19 at 18:34
  • $\begingroup$ By the way, you might be interested in my free differential geometry text. $\endgroup$ – Ted Shifrin May 13 '19 at 18:39
  • $\begingroup$ Thank you for the help. I will go through the book. $\endgroup$ – monk-E May 13 '19 at 21:24

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