Let $X$ be a Riemann surface of genus $g\ge 2$ and $q$ a holomorphic quadratic differential on $X$. The differential $q$ defines a flat metric with conical singularities on $X$: if $q=f(z)dz^2$ locally then the metric is $|f(z)|dzd\overline{z}$. Let $l$ be a trajectory for $q$ on $X$, i.e. a complete geodesic for the singular metric which does not meet any singular point and so is of constant direction on $X$. It is known (it is proven for example in the book by Strebel) that the image of $l$ is dense in a domain of $X$.
Let $\pi:(\widetilde{X},\widetilde{q})\rightarrow (X,q)$ be a locally diffeomorphic universal cover.
It is my understanding that a lift $\widetilde{l}$ of $l$ is made like in the following picture:
In particular $\widetilde{l}$ is proper ($\widetilde{X}$ is written as $\mathbb{H}^2$).
Now let $g:[0,1]\rightarrow X$ be an arc in $X$ such that $l$ intersects $g$ infinitely many times and in particular $l(0)\in g([0,1])$. Now let $l'(t):=l(t+t_0), t_0\neq 0$ (i.e. $l'$ is a reparametrization of $l$) such that $l'(0)\in g([0,1])$.
If I consider $\widetilde{g}$ a lift of $g$ then there are two lifts $\widetilde{l}$ and $\widetilde{l}'$ of $l$ and $l'$ respectively such that $\widetilde{l}(0)\in \widetilde{g}([0,1])$ and $\widetilde{l}'(0)\in \widetilde{g}([0,1])$
My question is: are the images of $\widetilde{l}$ and $\widetilde{l}'$ disjoint?
My guess is yes, they are made like in this picture:
Am I right?