The curvature and torsion: how to distinguish the shape of a curve 
Still curvature and torsion
To have uniqueness in the Fundamental Theorem of Curves, in case curvature may take zero values, it is necessary to fix the Frenet-Serret frame of the curve in the following situations:
(1) In every point that is an isolated zero of the curvature function;
(2) Whenever the curvature function coincides with the zero function in some interval \([a,b]\), it is necessary to fix the Frenet-Serret frame at \(b\).
Note that the curve in the interval \([a,b]\) is a line segment that has the direction of the tangent vector at \(a\). Note also that, since \(k(a)=0\), the Frenet-Serret frame at \(a\) is not well defined and therefore one has to consider the limit of \(T(\epsilon)\) when \(\epsilon\) goes to zero, where \(T(\epsilon)\) is the Frenet-Serret frame at time \((a-\epsilon)\).
Remark:
There are functions that have no isolated zeros but are not nul at any interval of its domain. These functions require some more complex "tools" for their study, so they are not covered in this work.
