### 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.

FUNDAMENTAL THEOREM OF CURVES:

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 \(\mathbb{R}^{3}\):

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:

(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).