A curve is said to be helix if its tangent line have a constant angle with a fixed direction. i.e. $\langle T(t),u\rangle$ is constant for some unit vector $u$.
I am trying to prove: a regular curve with $\kappa(t)>0$ is helix if and only if $\frac{\kappa}{\tau}$ is constant.
I can show the sufficient part： if $\alpha(t)$ is helix, then $\langle T'\!, u\rangle=0$ and therefore $\langle N,u\rangle=0$, which implies $\langle-\kappa(t)T(t)+\tau(t)B(t), u\rangle=0$. We then have $$ \frac{\kappa(t)}{\tau(t)} = \frac{\langle T(t),u\rangle}{\langle B(t),u\rangle}, $$ which is constant, since $\langle T(t),u\rangle,\langle B(t),u\rangle$ are constant.
But I have no idea in proving the reverse direction.