Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 2 / 25H / b

At a fixed initial parameter choose an orthogonal map $Q$ sending $\tilde{T}$ to $T$, $\tilde{N}$ to $N$ and $\tilde{B}$ to $-B$. Its determinant is $-1$. The transformed curve $Q\tilde{\alpha }$ has Frenet frame $(Q\tilde{T},Q\tilde{N},-Q\tilde{B})$, so its curvature is $k$ and its torsion is $-\tilde{\tau }=\tau$.

Both its frame and the frame of $\alpha$ solve the same linear Frenet system with the same initial data. The uniqueness theorem for a linear ordinary differential system with continuous coefficients makes the frames identical on the whole interval. In particular $Q{\tilde{\alpha }}^{\prime }={\alpha }^{\prime }$, so their difference is a constant vector $v$. ThusThe orientation-reversing rigid motion accounts for the reversal of torsion.