Let us assume that $\alpha(s)$ is a unit speed curve with $\kappa > 0$. I'm trying to find the vector function $w(s)$ such that

$$T' = w \times T,\quad N' = w \times N,\quad B' = w \times B.$$

I see that this is an application of Frenet Serret with $T' = \kappa\times N, N' = \frac{T'}{|T|}$, and $B' = \langle-\tau, N\rangle$ and I am just not seeing how to group these guys to get what I want, i.e., vector $w$. Do I need to use the def of an osculating plane or the right hand rule? Thanks

  • $\begingroup$ Write $w$ as a linear combination of $T,N,B$ and find the coefficients. $\endgroup$
    – PAD
    Oct 8 '12 at 9:20
  • 2
    $\begingroup$ I think $N'=T'/|T|$ should be wrong. Also other reported formulas seems to be wrong: $\kappa\times N$, but $\kappa$ is not a vector. $\endgroup$
    – enzotib
    Oct 8 '12 at 13:01

If you do the procedure in my comment you find that $w=\kappa B -\tau T$ where $\kappa$ is the curvature and $\tau$ is the torsion of $\alpha$.


From Frenet we have:

$\left[ \begin{array}{c} T' \\ N' \\ B' \end{array} \right] = \left[ \begin{array}{ccc} 0 & \kappa & 0 \\ -\kappa & 0 & \tau \\ 0 & -\tau & 0 \end{array} \right].\left[ \begin{array}{c} T \\ N \\ B \end{array} \right],$

where the curvature and torsion are respectively denoted $\kappa$ and $\tau.$

So from the first row from the Frenet we have, $T' = \kappa N$ and then crossing both sides by $T(s)$, we get $w(s) \times T = b N \times T + c B \times T = \kappa N,$ which only holds if $b = 0$ and $c = \kappa$.

From the second row of the Frenet frame we have, $N' = -\kappa T + \tau B$ and then crossing both sides by $N(s)$, we get, $w(s) \times N = a T \times N + c B \times N = -\kappa T + \tau B,$ which only holds if $a = \tau$ and $c = \kappa$.

Finally, the third row of the Frenet frame, $B' = -\tau N$ and then crossing both sides by $B(s)$, we get $w(s) \times B = a T \times B + b N \times B = - \tau N,$ which only holds if $a = \tau$ and $b=0.$


I just happen to have done this problem and here is a link to my solution:


The vector you are looking for a called the Darboux vector and is used to describe the rotation of a rigid body following the curve.

My conclusion seems to have an opposite sign with the answer from @PAD, but I am pretty sure I am correct as the problem on my book requested for the verification of a given solution.


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.