Proving the ratio of curvature and torsion is constant.

Suppose $(T,N,B)$ is a given Frenet-Serret frame for the curve and suppose $T\cdot u=\cos(\theta)$ for some constant vector $u$. It is sufficient to show $\tau/\kappa=\cot(\theta)$.

Differentiating $T\cdot u=\cos(\theta)$ yields $N\cdot u=0$. This implies $u=\cos(\theta)T+\sin(\theta)B$ because we can assume $u$ has unit length.

Differentiate this equation to obtain $0=\kappa cos(\theta)N-\tau \sin(\theta)N$ and so $\tau/\kappa=\cot(\theta)$.

To show the converse, first find a $\theta$ such that $\tau/\kappa=\cot(\theta)$ and work backwards through the proof above.