According to my notes, a differentiable ruled surface of $\mathbb R ^3$ is a 2-dimensional $C^k$ submanifold of $\mathbb R ^3$ that can be described as a union of straight lines. I'm working on some problems which seem all to imply that such a surface must possess a parametrization of the form: $$\varphi (s,t)= \gamma (s) + tv(s),$$ where $\gamma\colon I \to \mathbb R^3$ is a regular curve parametrized by arc lenght and $v\colon I \to \mathbb R ^3$ is a vector field along $\gamma$ such that $v$ and $\gamma '$ are linearly independent. In particular, $C=\gamma(I)$ is assumed to be a 1-manifold and the parametrization $\gamma$ is assumed to be periodic if $C$ is diffeomorfic to $S^1$, global if not.
My question is: does every connected ruled surface necessarily admit such a parametrization? How can I explicitly construct one (if possible)?