The Poincare’ model of hyperbolic geometry consists of the unit disk where the geodesics are arcs of circles perpendicular to the unit circle . It turns out that given two points there is a unique circle orthogonal to passing through and .
A nice way to prove the uniqueness of geodesics is to use the stereographic projection. Let be the unit sphere in and its north pole. The stereographic projection puts in one-to-one correspondence each point with the unique point that lies on the line through and . For example, the unit disk corresponds to the southern hemisphere on the sphere .
Another property is that the stereographic projection maps circles to “circles”, in the sense that circles on through the north pole are mapped to lines in and those that don’t go through are mapped to actual circles in .
Armed with these two facts we can now easily establish the uniqueness of hyperbolic geodesics. Given two points consider their images on the southern hemisphere of . It is clear that and do not lie on a vertical line. Therefore, there is a unique plane through and perpendicular to the -plane in , i.e. . The intersection of this plane with is a circle that is perpendicular to the equator of . Under the stereographic projection it will be mapped to a “circle” through the points and, by conformality, its image will be orthogonal to the image of the equator, i.e., (the equator is fixed by the projection).