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.
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 i-line 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.
The theorem also allows us to define the hyperbolic triangle ABC for any A,B,C in E.
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
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).
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
A standard trigonometric identity shows that this is equivalent to the earlier
main asymptotic page