The curvature and torsion: how to distinguish the shape of a curve

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 $$s$$.

Examples

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.

Examples