TITLE: Shortest path on hyperboloid
QUESTION [13 upvotes]: 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.

REPLY [2 votes]: You might want to look at A. Pressley, Differential Geometry, text problem 8.1, he calculates the geodesics for the hyperboloid of one sheet  \begin{equation} x^2+y^2-z^2=1 \end{equation} and finds that they are actually four, two straight lines, one circle and one hyperbola. But he keeps things very simple and finds them just by using some basic definition.