everyone. Let $S$ be a closed orientable 2-surface with genus at least 2. Let $\{C_{i}\mid i=1,...,3g-3\}$ be a pants decomposition of $S$. Suppose that $\{l_{C_{i}},\tau_{C_{i}}, for~ i=1,...,3g-3\}$ be a point in Fenchel Nielsen coordinates. Then $S$ admits a hyperbolic metric with respect to the point.

Now I want to know:

For any given integer $n\geq 2$, is there a way to classify any two geodesics with distance at least $n$ from the geometry, such as the length of these two geodesics?

Note: The distance defined on two closed geodesics is the distance in curve complex.

And I know that using pseudo-anosov map is a good way to construct distance at least
$n$ two geodesics. But I really want to know is it possible to get this just by using the
metric defined on the surface.