The Fundamental Theorem of Curves

In the planar case there is one more concept: the curvature (with sign). In this case, a plane curve is determined by its curvature with sign, up to some euclidean motion. This fact suggests a similar fact for space curves: a space curve is determined by its curvature and torsion, up to some euclidean motion. This is indeed true but requires an additional condition to ensure uniqueness.


Let \(P_{0}\) in \(\mathbb{R}^{3}\) and let \(TF_{0}\) be an orthonormal frame with positive orientation. Further, let \(k,\tau:\, I\rightarrow\mathbb{R}^{3}\) be continuous functions, with \(k>0\) in \(I\).

Then, there is one, and only one, \(C^{2}\) curve, \(c:\, I\rightarrow\mathbb{R}^{3}\), parametrized by arc length, whose curvature and torsion are precisely \(k\) and \(\tau\), whose starting point is \(P_{0}\) and whose initial Frenet-Serret frame is \(TF_{0}\).

(See, for example: M. Spivak, A comprehensive introduction to differential geometry (vol. 2); 1999)

Observations: The assumption that curvature is strictly positive is essential to guarantee the uniqueness of the curve. In fact:

(1) Consider the following functions in \(\mathbb{R}^{3}\):

\[f(t)=\left(t^{3},t,0\right);\] \[g(t)=\begin{cases} \left(t^{3},t,0\right), & \mbox{if }t<0\\ \left(0,t,t^{3}\right), & \mbox{if }t\geq0 \end{cases}.\]

The two functions have the same curvature, which is zero in \(t=0\). The torsion in both cases is zero at every point where is defined (that is, all points except \(t=0\)). The initial point and the initial Frenet-Serret frame also coincide in the two curves (note that the curves coincide until instant \(t=0\)). But there is no Euclidean motion that takes one curve to the other, as is shown in the following picture:

(2) Another example similar to the previous one:

\[f(t)=\begin{cases} \left(0,0,0\right), & \mbox{if }t=0\\ \left(t,0,5e^{-t^{-2}}\right), & \mbox{if }t\neq0 \end{cases};\] \[g(t)=\begin{cases} \left(t,5e^{-t^{-2}},0\right), & \mbox{if }t<0\\ \left(0,0,0\right), & \mbox{if }t=0\\ \left(t,0,5e^{-t^{-2}}\right), & \mbox{if }t>0 \end{cases}.\]

(A. Gray, Modern Geometry of Curves and Surfaces, pp. 142-145)

Such kind of examples justifies the option for not considering that the torsion is zero when the curvature is zero. With this option we still have that a curve has null torsion in an interval if and only if the curve is planar in that interval, otherwise not (recall, for example, functions \(g\) presented above).