Does the velocity vector have to have unit speed parametrisation in calculations of the frenet frame?

Question

Consider the curvature of a curve $\beta$ at a point s.

This is given by $\kappa(s):=|T'(s)|$, where $T(s)=\beta '(s) $.

similarly we define the fields in the frenet frame $\{T,N,B\}$ by

$$T(s)=\beta'(s)$$

$$N(s):=\tfrac{T'(s)}{\kappa(s)}$$

$$B(s):=T(s)\times N(s)$$

My question is does $\beta'(s)$ have to be parametrised to unit speed in all of these calculations ?

Answer 1

Yes. More precisely, these formulas only work for curves parametrirized by the arclength. Otherwise, you would not have, for instance, that $\bigl\lVert T(s)\bigr\rVert=1$.