The curvature and torsion: how to distinguish the shape of a curve 
Still curvature and torsion (III)
In order to have the same planar curves, both in two-dimensional case and in the three-dimensional case, we "allow" in the Fundamental Theorem of Curves that the curvature may take zero values. To ensure the uniqueness of the curve, this requires, besides the choice of the curvature and torsion functions, the starting point and the starting Frenet-Serret frame, the choice of the Frenet-Serret frame at some more instants, namely the ones where \(k=0\).
For simplicity, we study for now only the case where the curvature function has only isolated zeros.
Try to find out experimentally, with the help of 3D app (\(k\geq 0\), finite number of zeros), in which instants it is needed to choose the Frenet-Serret frame. Observe the corresponding changes in the curve when those frames are changed.
As you may have noticed, depending on the choice of the Frenet-Serret frames that appear in the left side of the app, there exist different (\(C^{2}\)) curves with the same curvature and torsion (recall that the torsion is defined only at points where the curvature is not zero).
Therefore, to guarantee the uniqueness of the curve, it is really necessary to assign the Frenet-Serret frame in certain instants.
Then a new problem arises: the curves with torsion zero (in the points where torsion is defined) are not always plane curves. Search on the 3D app (\(k\geq 0\), finite number of zeros) for examples of such curves.
In any case, note that for a curve to be plane it is necessary that torsion is zero at all points where it is defined (the problem is that this condition is not sufficient to ensure that the curve is actually in a plane).
But after all, in case we need to state Frenet-Serret frames in order to get the uniqueness of a three-dimensional curve that has zero torsion (in points where it is defined), how many choices can we make in order that the curve is, indeed, a plane curve? Try to find experimentally the answer to this question with the help of the 3D app (\(k\geq 0\), finite number of zeros).
And if the zeros of the curvature function are not all isolated points?
