In this section, we establish basic results about hyperbolic lines and angles.
This includes results illustrated in the experimental pages.
Taking the kleinian view, we base our development of the theory
Suppose that H is an h-line.
We refer to the restriction of iL to D as h-inversion with respect to H, denoted by iH.
The h-inversions play a role in hyperbolic geometry very similar to the role
Theorem 1 If t is a hyperbolic transformation, then
In our Poincare disk model, O appears to have a special status, since
the h-lines through O are euclidean segments rather than arcs of circles.
In fact, it is typical, as we shall see.
The CabriJava pane on the right illustrates the Lemma.
More generally, we have the
Since the h-inversion, and hence the h-line, in the Interchange Lemma are unique, we make the
The justification for the names h-bisector and h-midpoint will become clear
These Lemmas can be used to prove fundamental results on
The Incidence Theorem
Note that these are exact analogues of incidence results in
There are many analogues of euclidean results. We give some
The Angle Bisector Theorem
This is left as an exercise to the reader. Simply apply the Origin Lemma
The Perpendicular Theorem
If H is an h-line, then, for any point P,
there is a unique h-line through P perpendicular to H.
In the figure, you can move the point P,
In euclidean geometry, we say that a figure P is
symmetric about a line L if reflection in L maps P to itself.
Symmetry of h-lines
This is a consequence of the Mirror Property from inversive geometry.
We shall meet more interesting examples when we look at hyperbolic