TITLE: Geodesic Uniqueness in the Hyperbolic Plane
QUESTION [8 upvotes]: I am studying Hyperbolic Geometry. At this part, I have proved that semicircles and straight lines orthogonals to the real axis are geodesics in the hyperbolic plane. But how I proof that this geodesics are uniques? That it does not exist others geometric places between two different points that minimizes the hyperbolic length? 
I have seen a topic related to this, but I do not know nothing about Riemannian Geometry. If someone could indicate to me some references to this proof(or the idea behind) I will be very grateful!

REPLY [7 votes]: You can argue "geometrically" as follows: For two points $p$, $q$ on the imaginary axis it is pretty obvious that the segment $[p,q]$ is the unique shortest curve connecting $p$ and $q$. Now if $p$, $q\in H$ are arbitrary there is a Moebius transformation $T: H\to H$ sending $p$ and $q$ to two points $p'$, $q'$ on the imaginary axis. Since $T$ is a hyperbolic isometry it follows that $p$ and $q$ have a unique shortest connection, and since $T$ (now considered as a map $\bar {\mathbb C}\to \bar {\mathbb C}$) maps circles to circles and preserves angles between curves it follows that this shortest connection lies on a circle intersecting the real axis orthogonally.