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I'm trying to find the curvature and torsion of the tangent indicatrix of a curve with respect to the curvature and torsion of the initial curve, that is, if $\kappa, \tau$ are the curvature and torsion of $\alpha(s)$, what is the curvature and torsion of $\beta(s) = \alpha'(s)$?
I'm aware of The curvature and torsion of the tangent indicatrix
However, I don't understand how do you get $\sigma'\times\sigma''=\kappa^3B+\kappa^2\tau T$

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Actually, it's simple: $$\sigma' = \kappa N$$ $$\sigma'' = \kappa'N-\kappa^2T+\kappa\tau B$$ Thus $$\sigma' \times \sigma'' = \kappa^3B+\kappa^2\tau T$$

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