ATRACTOR

The osculating plane at a given instant is defined by the tangent and the normal vectors of Frenet-Serret frame at that same instant (note that this plane is perpendicular to the binormal vector).

If torsion is positive, the curve "turns" to the side to which binormal vector points. If torsion is negative, the curve "turns" to the opposite side. A curve "can make this choice" at a given instant, only if its binormal vector at that instant is well defined. In fact, by the construction of Frenet-Serret frame, the binormal is defined only if the normal vector is defined (that is, in the points where curvature is greater than 0).

Note that when the torsion in some point is nonzero, the curve "goes out" of its osculating plane (in that point) and, therefore, a tridimensional curve, defined in an interval, is planar if and only if its torsion is zero (note that in the latter fact we are assuming that torsion is everywhere defined in the interval and that only happens if curvature is always greater than 0).