Ideal point

In hyperbolic geometry, an ideal point, omega point or point at infinity is a well defined point outside the hyperbolic plane or space.
Given a line "l" and a point "P" not on "l", right- and left-limiting parallels to "l" through "P" converge to "l" at "ideal points".
Unlike the projective case, ideal points form a boundary, not a submanifold. So, these lines do not "intersect" at an ideal point and such points, although well defined, do not belong to the hyperbolic space itself.
The ideal points together form the Cayley absolute or boundary of a hyperbolic geometry. 
For instance, the unit circle forms the Cayley absolute of the Poincaré disk model and the Klein disk model.
While the real line forms the Cayley absolute of the Poincaré half-plane model .
Pasch's axiom and the exterior angle theorem still hold for an omega triangle, defined by two points in hyperbolic space and an omega point.
Polygons with ideal vertices.
Ideal triangles.
if all vertices of a triangle are ideal points the triangle is an ideal triangle.
Ideal triangles have a number of interesting properties:
Ideal quadrilaterals.
if all vertices of a quadrilateral are ideal points the quadrilateral is an ideal quadrilateral.
While all ideal triangles are congruent, not all quadrilaterals are, the diagonals can make different angles with each other resulting in noncongruent quadrilaterals
having said this:
Ideal square.
The ideal quadrilateral where the two diagonals are perpendicular to each other form an ideal square.
It was use by Ferdinand Karl Schweikart in his memorandum on what he called "astral geometry", one of the first publications acknowledging the possibility of hyperbolic geometry.
Ideal "n"-gons.
An ideal "n"-gon can be subdivided into ideal triangles, with area times the area of an ideal triangle.
Representations in models of hyperbolic geometry.
In the Klein disk model and the Poincaré disk model of the hyperbolic plane. In both disk models the ideal points are on the unit circle (hyperbolic plane) or unit sphere (higher dimensions) which is the unreachable boundary of the hyperbolic plane. 
When projecting the same hyperbolic line to the Klein disk model and the Poincaré disk model both lines go through the same two ideal points.(the ideal points in both models are on the same spot).
Klein disk model.
Given two distinct points "p" and "q" in the open unit disk the unique straight line connecting them intersects the unit circle in two ideal points, "a" and "b", labeled so that the points are, in order, "a", "p", "q", "b" so that |aq| > |ap| and |pb| > |qb|. Then the hyperbolic distance between "p" and "q" is expressed as
Poincaré disk model.
Given two distinct points "p" and "q" in the open unit disk then the unique circle arc orthogonal to the boundary connecting them intersects the unit circle in two ideal points, "a" and "b", labeled so that the points are, in order, "a", "p", "q", "b" so that |aq| > |ap| and |pb| > |qb|. Then the hyperbolic distance between "p" and "q" is expressed as
Where the distances are measured along the (straight line) segments aq, ap, pb and qb.
Poincaré half-plane model.
In the Poincaré half-plane model the ideal points are the points on the boundary axis. 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).
Hyperboloid model.
In the hyperboloid model there are no ideal points.