boundary at infinity of $\mathbb{H}^2$

Question

In hyperbolic geometry what does it mean when they say the boundary at infinity of $\mathbb{H}^2$? The only idea I came up with was a horizontal line to represent the horizon and to lines meeting at a point on that line, but they are not actually intersecting at the horizon line. The road goes off to infinity. Or do I have the wrong idea here?

Answer 1

Independent of any model, we may say that the points at $\infty$ are the equivalence classes of half-lines in $\mathbb H$ under the relation $\sim$, where $L\sim L'$ if and only if as $p\in L$ goes to infinity (away from the endpoint), the distance from $p$ to $L'$ goes to $0$. As an example, in the half-plane model, two vertical half-lines are similar under $\sim$.