The curvature and torsion: how to distinguish the shape of a curve 
Still curvature and torsion (II)
The fact that the curvature can not take the zero value in the Fundamental Theorem of Curves is a drawback.
Firstly, observe that there are curves in the 2D app (obviously planar) that are not constructed from the 3D app (\(k>0\)). Try to find some of them.
As you may have noticed, the problem arises when the curvature of the curves in the app 2D takes positive and negative values. In this situation the curve does not turn always to the same side (it "turns" to the left at the points where the curvature is positive and it "turns" to the right at the points where the curvature is negative) while the planar curves considered in the three-dimensional case "turn" always to the same side.
As you may have noticed yet, for a curve to be planar, the torsion must be always zero.
