### Conclusion

A plane curve is usually represented by an equation involving two coordinates \(x \) and \(y \). It is then possible to describe its curvature with a formula. However, alternatively, it is also possible, as we have seen, to take the curvature as a primitive notion and to express the curve in a more natural way. The idea is not to think about the relative positions of the curve points with respect to a coordinate system \(OXY\) but instead to think on the curve marked by units of "arc length" where arc length is a length measured along the curve (as if it were stretched rectilinearly along a ruler). It is then natural to describe the curve with an equation \(k=\gamma(s)\) that specifies the curvature \(k\) in terms of the arc length \(s\).

We have also observed how to do this in the three-dimensional case, again forgetting about a fixed coordinate system \(OXYZ\) and replacing it by a mobile coordinate system, the Frenet-Serret frame \(\left\{T,N,B\right\}\), which moves along the curve, adapting to it.

It is possible to define, at each point of the curve, two numerical quantities,
called *curvature* and *torsion*, that allow a full description of
the curve. The curvature of a curve in three-dimensional space is essentially
the same as the planar case: it measures the tendency of the curve at each point
to move away from a straight line (specifically, the tangent line to the curve
at that point). The torsion measures the tendency of the curve to move away
from a plane. Imagining the Frenet-Serret frame moving along the curve, the
curvature measures the tangent variation rate \(T\) relative to a fixed direction
(i.e., it is given by the derivative of the module \(T\)) while torsion measures
the change rate of the binormal \(B\).

As in the planar case, a space curve can be completely specified by curvature and torsion functions (as functions of the arc length). These equations have the form \(k=\gamma(s)\) and \(\tau=\delta(s)\). As we have seen, the size and shape of the curve are uniquely determined by these equations.