On the sphere $S^2$, the shortest path between two points is the great circle path. How about $H^2$, the hyperboloid $x^2+y^2-z^2=-1, z\ge 1$, with the Euclidean distance? Is there a formula for the shortest path between two points on the surface? And what is the length of the shortest path?
Note that it is not the hyperbolic distance; it is the Euclidean distance.