I am soon going to write an exam in differential geometry. A typical task will most likely be to find the curvature of a given parameterized curve.

However, there are different ways to find the curvature of such curves. Let's assume we have a three-dimensional curve $$\gamma(t):I \to \mathbb{R}^3$$. If the given curve is parameterized by arc length, we simply find

$$ \kappa(t) = \ddot{\gamma}(t) $$

If the given curve is not parameterized by arc length however, we have to use the general formula for a curve given by

$$ \kappa (t)={\frac {|\dot{\gamma}(t)\times \ddot{\gamma}(t)|}{|\dot{\gamma(t)}|^3}}$$

My actual question now is what you would suggest to be the best approach to find the curvature of any given curve in an exam where time matters a lot.

During my homework i have plenty of time and i don't mind checking whether or not a given curve is already parameterized by arc length or even parameterizing it by arc length myself. During an exam however, this approach is most likely going to fail due to the limited time we have.

I'd like to know if i should always use the general formula to find the curvature of any given curve or what would be the "ideal way" to approach tasks like this in general.

Thank you very much. Please excuse me for any mistakes i might have done in grammar or orthography.

  • $\begingroup$ Arc-length parametrized curves very rarely have analytical closed-form expressions, so if given a curve parametrized by an analytical formula, I would assume it's almost certainly not parametrized by arc length (unless it is clearly something simple like a line, circle, or helix). $\endgroup$ – Rahul Apr 10 '18 at 14:55
  • $\begingroup$ @Rahul, thank you very much! I guess this will also be the case in my exam. I will stick to the general formulas and practise applying them. $\endgroup$ – Zest Apr 10 '18 at 16:18

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