We know that any $2$-dimensional manifold $M$ with constant negative curvature is locally isometric to hyperbolic plane $\mathbb{H}_2$. What happens to the inverse image, under the isometry, of the geodesic $\gamma$ that connects a point $p\in \mathbb{H}_2$ and another point $q\in \partial \mathbb{H}_2$ at infinity? But since $q\notin \mathbb{H}_2$, i.e ideal points do not belong to $\mathbb{H}_2$, I doubted whether this is meaningful.