Computing $T'\times T''$ in terms of the Darboux vector.

Question

I'm trying to show that $T ' \times T ''= kw$ where $k$ is curvature and $w$ is the Darboux vector.

I know $T' = w \times T$. How should I proceed?

Answer 1

Hint: first write $w=aT+bN+cB$ and use the definition of the Darboux vector (those three equations, yes) to solve for $a$, $b$ and $c$ in terms of the curvature and torsion of the curve. Then use the Frenet equations to compute $T'\times T''$ and recognize in the result the expression for $w$ you previously found.

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A further hint: since $(T,N,B)$ is a positive orthonormal basis, the following relations hold: $$T\times N=B,\qquad N\times B=T,\qquad B\times T=N. $$