# Calculating curvature with arc length parameterizatoin versus non arc length parameterization

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?

Thanks!