But this is the well-known line element of hyperbolic three-space H in the upper half space chart. This is closely analogous to the well known upper half plane chart for the hyperbolic plane H, which is familiar to generations of complex analysis students in connection with conformal mapping problems (and much more), and many mathematically minded readers already know that the geodesics of H in the upper half plane model are simply semicircles (orthogonal to the circle at infinity represented by the real axis).