1
$\begingroup$

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 ?

$\endgroup$
1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.