Computing the curvature and torsion in any parametrisation
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In the lectures on curvature and torsion, we defined the curvature and torsion using an arclength parametrisation. We do this because the arclength parametrisation is a canonical parametrisation for the curve, and the curvature and torsion are intrinsic quantities associated to the curve.
Unfortunately, we have seen that reparametrising by arclength is often complicated, even if the parametrisation is very simple (recall that we encountered this when computing the osculating circle for the parabola and the cubic curve). First we have to find the inverse of
\[
s(t) = \int_{t_0}^t \left| p'(u) \right|\, du .
\]
The resulting formula is often quite complicated, and we then have to take derivatives to compute the components of the Frenet frame, from which we can compute the curvature and torsion. These issues make it difficult to compute the curvature and torsion in practice. Today we will write down some formulas for curvature and torsion in any parametrisation, which will then make it easier to compute these quantities directly, instead of having to reparametrise by arclength.
The following result is proved in Theorem 1.13 in the Lecture Notes.
Theorem. (Curvature and torsion in any regular parametrisation)
Let \(p : I \rightarrow \mathbb{R}^3\) be a smooth regular curve. Then
\[
\begin{aligned}
k(t) & = \frac{1}{|p'|^2} \left( p'' - \frac{p' \cdot p''}{|p'|^2} p' \right) \\
\kappa(t) & = \frac{|p' \times p''|}{|p'|^3}.
\end{aligned}
\]
If the point \(p(s)\) is not an inflection point, then
\[
\tau(t) = -\frac{1}{\kappa(t)^2} \frac{[p',p'',p''']}{|p'|^6} = - \frac{[p',p'',p''']}{|p' \times p''|^2} .
\]
Exercise.
Now that we have these formulas in hand, you can revisit the examples of the parabola and the cubic curves and compute the curvature in both cases to see that it agrees with the formulas stated in these examples.
Next time
We will prove the Frenet formulas which show that the curvature and torsion determine a differential equation for the Frenet frame.
In Problem Class 2 we will use the Frenet formulas to give another proof of the above formulas for curvature and torsion.
