Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question
    
Write $w$ as a linear combination of $T,N,B$ and find the coefficients. –  PAD Oct 8 '12 at 9:20
2  
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. –  enzotib Oct 8 '12 at 13:01
add comment

2 Answers

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 $\alpha$.

share|improve this answer
add comment

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 $b=0.$

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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