Similarly to this question, I don't really get the way lines of striction are introduced in Differential geometry of curves and surfaces.
Specifically, it is stated that a curve $\beta(t)$ being contained in the trace of a ruled surface parametrized by $x : I\times \mathbb{R}$, $x(t, v) =\alpha(t) + vw(t)$ is equivalent to the existance of some $u(t)$ such that
\begin{align*} \beta(t) = \alpha(t) + u(t)w(t) \end{align*}
I understand that this is required in this case, since we eventualy want $\beta$ to be a directrix of this surface. But in general wouldn't curves such as
\begin{align*} \gamma(v) = x(t_0, v) \end{align*}
for fixed $t_0 \in I$ contradict this requirement?