Let $I$ be an open interval, $\gamma:I \rightarrow \mathbb{R}^3$ be a curve which is parameterized by arc length and $\gamma(s)$ is in unit sphere for all $s\in I$, and $\tau$ be the torsion of $\gamma$.
Now, I want to prove below;
If $\tau(s)=1$ for all $s\in I$, then there exists some $a,b\in\mathbb{R}$ such that $\displaystyle{\kappa(s)=\frac{1}{a\sin s+b\cos s}}$. Note that $\kappa(s)$ is curvature of $\gamma$.
I try to prove it by calculating according to the definition, but formula is too complicated and doesn't work well...