Let $\alpha(t)$ be a non arc-length parameterized regular curve. I understand that guess you want to get the curvature from taking the derivative of a unit vector, so would it hold then that $\kappa(t)= |\frac{d}{dt}\frac{\alpha'(t)}{|\alpha'(t)|}|$ for non-arc length parameterized curves?

Could somebody help reconcile this with the frenet equations?

$N \cdot \kappa(t)=T'$

Where this equation, when $\alpha(t)$ is not arc length parameterized, will have an extra term, $v(t)=s'(t)$ and then:

$N \cdot \kappa(t)s'(t)=T'$

But yeah, my main question is I'm wondering if this is true: $\kappa(t)= |\frac{d}{dt}\frac{\alpha'(t)}{|\alpha'(t)|}|$ for non arc length parameterized curves, and if not, why not?



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