## Uniqueness of hyperbolic geodesics

The Poincare’ model of hyperbolic geometry consists of the unit disk $\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}$ where the geodesics are arcs of circles perpendicular to the unit circle $\partial\mathbb{D}=\{|z|=1\}$. It turns out that given two points $z,w\in\mathbb{D}$ there is a unique circle orthogonal to $\partial\mathbb{D}$ passing through $z$ and $w$.

A nice way to prove the uniqueness of geodesics is to use the stereographic projection. Let $\mathcal{S}$ be the unit sphere in $\mathbb{R}^3$ and $N$ its north pole. The stereographic projection puts in one-to-one correspondence each point $z\in\mathbb{C}$ with the unique point $z^*\in \mathcal{S}\setminus\{N\}$ that lies on the line through $N$ and $z$. For example, the unit disk $\mathbb{D}$ corresponds to the southern hemisphere on the sphere $\mathcal{S}$.

One of the main properties of the stereographic projection is that it is a conformal map, i.e., it preserves angles between curves. Proving this is a nice exercise in geometry.

Another property is that the stereographic projection maps circles to “circles”, in the sense that circles on $\mathcal{S}$ through the north pole are mapped to lines in $\mathbb{C}$ and those that don’t go through $N$ are mapped to actual circles in $\mathbb{C}$.

Armed with these two facts we can now easily establish the uniqueness of hyperbolic geodesics. Given two points $z,w\in\mathbb{D}$ consider their images $z^*, w^*$ on the southern hemisphere of $\mathcal{S}$. It is clear that $z^*$ and $w^*$ do not lie on a vertical line. Therefore, there is a unique plane through $z^*$ and $w^*$ perpendicular to the $(x,y)$-plane in $\mathbb{R}^3$, i.e. $\mathbb{C}$. The intersection of this plane with $\mathcal{S}$ is a circle that is perpendicular to the equator of $\mathcal{S}$. Under the stereographic projection it will be mapped to a “circle” through the points $z,w$ and, by conformality, its image will be orthogonal to the image of the equator, i.e., $\partial\mathbb{D}$ (the equator is fixed by the projection).