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If $t(t),n(t),b(t)$ are rotating, right-handed frame of orthogonal unit vectors. Show that there exists a vector $w(t)$ (the rotation vector) such that $\dot{t} = w \times t$, $\dot{n} = w \times n$, and $\dot{b} = w \times b$

So I'm thinking this is related to Frenet-Serret Equations and that the matrix of coefficient for $\dot{t}, \dot{n}, \dot{b}$ with respect to $t,n,b$ is skew-symmetric.

Thanks!

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1 Answer 1

up vote 4 down vote accepted

You have sufficient information to compute it yourself! :)

Suppose that $w=aT+bN+cB$, with $a$, $b$ and $c$ some functions. Then you want, for example, that $$\kappa N = T' = w\times T = (aT+bN+cB)\times T=b N\times T+cB\times T=-bB+cN.$$ Since $\{T,N,B\}$ is an basis, this gives you some information about the coefficients. Can you finish?

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@ Mariano: I'm not very familiar with linear algebra. What does basis have to do with the coefficients? –  Lindsay Duran Sep 23 '11 at 19:01
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@LindsayDuran, you should really familiarize yourself with linear algebra: explaining this is a bit outside of the scope of comments! –  Mariano Suárez-Alvarez Sep 23 '11 at 19:02
    
@Mariano: I got $\omega$ in terms of $\tau$ and $\kappa$, but the problem didnt specify a curve. How should I deal with this? –  ninja Sep 23 '11 at 20:00
    
@ninja: That's fine, as both curvature and torsion should indeed be involved. What else were you expecting? –  J. M. Sep 24 '11 at 3:04

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