In the disk model, the hyperbolic plane consists of the interior of the unit circle. The points of the boundary C do not belong to the plane. As we shall see, these boundary points play a role similar to that of the "points at infinity" in euclidean geometry. In different texts, these are known as the asymptotic, ideal, limit or omega points of the geometry. We shall use the first of these.
definition
When we need to make it clear that we are dealing with a point of D, we shall
A hyperbolic line H is the intersection of D with an iline H* orthogonal to C. The
When we wish to emphasise that the asymptotic points are included, we shall Likewise, a hypercircle "has" two asymptotic points, and a horocycle one.
A hyperbolic transformation h is the restriction to D of an inversive transformation Note. We have looked at the action of H(2) on C by itself in weird geometry.
Theorem HA1
proof
definitions
The theorem also allows us to define the hyperbolic triangle ABC for any A,B,C in E.
definition
Observe that, if A is an asymptotic point, the hyperbolic lines AB, AC are
The sketch shows a trebly asymptotic triangle ABC, two doubly asymptotic
Theorem HA2
For triangles which are singly or doubly asymptotic this can be proved by
Since asymptotic triangles have some sides of infinite length, we cannot


The angle of parallelism If ABC is a singly asymptotic triangle, with A on C, d(B,C) = l, <ABC =½π, <ACB = γ, then tan(γ) = 1/sinh(l), i.e. sinh(l) = cot(γ).
proof This may be viewed as the asymptotic analogue of the tangent formula.
Using standard formulae from trigonometry and hyperbolic trigonometry,
As an exercise, the reader may verify that the result may be restated as
The same limiting argument for a general singly asymptotic triangle when
Theorem HA3
A standard trigonometric identity shows that this is equivalent to the earlier 
