Are points on the boundary of the Poincare Disc Model considered part of the hyperbolic plane?

Question

Are the points that lie on the outside edge of a Poincare Disc considered to be in the hyperbolic plane? In other words, are points C, D, J, Y, and Z inside the plane?

Answer 1

No, the perimeter is not part of the model.

The lines that approach a mutual point in the perimeter are considered parallel.

In hyperbolic geometry, two distinct parallel lines are further classified as one of two types of parallel

• limit parallel: if the two lines approach a common point on the perimeter of the circle
• ultraparallel: if they are not limit parallel.

In your picture, the line through $AB$ and the line through $KL$ are ultraparallel, while $EF$ is limit parallel to $AB$.

Answer 2

No, they are not elements of the hyperbolic plane. The boundary circle is sometimes called "ideal boundary" because of that. Its elements can be thought of as equivalence classes of geodesic rays: two such rays are equivalent (i.e., go to the same ideal boundary point) if they stay at bounded distance from each other when parametrized by arclength.

In intuitive terms, it's like points of the horizon vs points on the Earth's surface.

Using the aforementioned idea of geodesic rays, one can give the ideal boundary more structure: see Wikipedia.