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

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 ?

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