Ideal point

Unlike the projective case, ideal points form a boundary, not a submanifold.

The ideal points together form the Cayley absolute or boundary of a hyperbolic geometry.

The real line forms the Cayley absolute of the Poincaré half-plane model.

In the Klein disk model and the Poincaré disk model of the hyperbolic plane the ideal points are on the unit circle (hyperbolic plane) or unit sphere (higher dimensions) which is the unreachable boundary of the hyperbolic plane.

There is also another ideal point that is not represented in the half-plane model (but rays parallel to the positive y-axis approach it).