Vector calculus, just learned about the Frenet frame and curvature and torsion. Naturally, we have to calculate a lot of these on homework and exams. However, the formulas that we are given for getting curvature, torsion, frame are computationally intensive and usually requires a whole bunch of different calculations (differentiate twice, take a cross product and two absolute values just for $\kappa$ and $\tau$). Being unreliable as a biological computer, I am fairly error prone. It feels like there should be easier and more direct ways of getting these formulae. We have:
$\mathrm T=\frac{\mathrm r^\prime}{|r^\prime|}$, $\mathrm B=\frac{\mathrm{r'(\mathit t)\times r''(\mathit t)}}{|\mathrm{r'(\mathit t)\times r''(\mathit t)}|}$, $\mathrm{N=B\times T}$, $\kappa =\frac{|\mathrm{r'(\mathit t)\times r''(\mathit t)}|}{\mathrm |r'(t)|^3}$, $\tau=\frac{(\mathrm{r'(\mathit t)\times r''(\mathit t)})\cdot \mathrm r'''(t)}{|\mathrm{r'(\mathit t)\times r''(\mathit t)}|^2}$
and the Frenet-Serret formulas.
I am not sure what strategy I should go for – calculate the unit tangent, differentiate a whole bunch, and take absolute values (go straight for Frennet-Serret) or should I try the given formulas since I have a function with respect to t and not arc length? Is there some other better faster way to do this?
Best Answer
The sequence you suggest is pretty minimal. I tell students to avoid finding $\mathbf{N}$ by finding $\mathbf{T}^\prime/|\mathbf{T}^\prime|$ since this computation can be rather involved. Using $\mathbf N=\mathbf B\times \mathbf T$ is usually cleaner.