Still curvature and torsion

As we shall see, curvature and torsion are enough to describe the shape of any tridimensional curve.

The curvature measures, at each point, how rapidly the curve deviates from its tangent line at that same point.  Important observation about curvature...

The torsion measures, at each point, how rapidly the curve deviates from its osculating plane at that same point.  Important observation about torsion...

Are curvature and torsion enough information to define the shape of a curve? In fact, there is a similar result to the Fundamental Theorem of Plane Curves in the tridimensional case, but it is necessary to add an additional constraint to the curvature function: it is always greater than zero. This assumption is essential for obtaining the uniqueness of the curve. We then have the Fundamental Theorem of Curves, which ensures that if two curves have the same curvature function (with \(k>0\)) and the same torsion, there is a rigid motion in \(\mathbb{R}^{3}\) that applies one curve into the other. This means that it is possible to transform each of the two curves into the other with no deformation, that is, by means only of translations and rotations.

Hence, any curve is uniquely defined by four invariants

  • the curvature (\(>0\));
  • the torsion
  • the initial point of the curve (it corresponds to the translation);
  • the initial Frenet-Serret frame of the curve (it corresponds to the rotation).

See the following applet to confirm that.

What can we conclude in the case that the curvature function is not always greater than zero?