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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

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    Write $w$ as a linear combination of $T,N,B$ and find the coefficients.2012-10-08
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    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.2012-10-08

3 Answers 3