Oriented geodesic rays, lines, parameters and tails

In a geodesic metric space (Geodesics and geodesic metric spaces), an oriented geodesic ray is an isometric embedding $\rho :[0,\mathrm{\infty })\to X$. Its origin is $\rho (0)$, its parameter increases in the chosen orientation, and its $T$-tail is $\rho ([T,\mathrm{\infty }))$, for $T\ge 0$.

An oriented geodesic line is an isometric embedding $\mathrm{\ell }:\mathbb{R}\to X$. Its origin is $\mathrm{\ell }(0)$ and its signed parameter increases in the chosen orientation. Its positive and negative $T$-tails are $\mathrm{\ell }([T,\mathrm{\infty }))$ and $\mathrm{\ell }((-\mathrm{\infty },-T])$. Replacing $\mathrm{\ell }(t)$ by $\mathrm{\ell }(-t)$ reverses the orientation.

The maps include parameter and origin data; their images alone do not. Two rays have a common tail when $\rho ([S,\mathrm{\infty }))=\sigma ([T,\mathrm{\infty }))$ for some specified nonnegative $S,T$. Hausdorff distances between rays or lines always refer to their images.