Simple question about curvature on planar curves

Question

So, I am studying Differential Geometry in university and i was given the following definition: \begin{equation*} K(s) = ||\beta''(s)|| \end{equation*} Where $\beta(s)$ is a parametrization of a curve such that $||\beta'(s)|| = 1, \forall s$, and where $K(s)$ denotates the curvature of the curve.

I was wondering if this result is ALWAYS valid, including for planar curves. I know that we can "associate" a signal to curvature in planar curves, but my teacher solved an exercise involving a planar curve purely through the definition above, and i began to wonder if i can always do it through definition, independent of the planar curve in study.

Thanks for all the help in advance guys,

Best regards.

Answer 1

Yes, curvature is always defined thus. For plane curve there is a signed curvature, see: https://en.wikipedia.org/wiki/Curvature#Plane_curves, but it is defined in terms of the unsigned quantity.