Hyperbolic inversions

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
on a group of transformations of the hyperbolic plane.

Suppose that H is an h-line.
Then H = LnD, where L is an i-line orthogonal to C.
From the Mirror Property, inversion with respect to L, iL, maps D to D.

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
of reflections in euclidean geometry. For example:
The h-inversions generate the hyperbolic group H(2) whose elements are
called hyperbolic transformations.

Theorem 1 If t is a hyperbolic transformation, then
  • t maps h-lines to h-lines,
  • t preserves angles between h-lines.
Proof of Theorem 1

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.

Origin Lemma
For any point P ε D, there is an h-inversion mapping P to O and O to P.
If P ≠ 0, then the h-inversion is unique.

Proof of the Origin Lemma

The CabriJava pane on the right illustrates the Lemma.
You can move P and note how the green h-line varies.
Of course, if you drag P outside the circle, the is no h-line.

More generally, we have the

Interchange Lemma
If P and Q ε D, there is an h-inversion mapping P to Q and Q to P.
If P ≠ Q, then the h-inversion is unique.

Proof of the Interchange Lemma.

Since the h-inversion, and hence the h-line, in the Interchange Lemma are unique, we make the

If P and Q are distinct points of D, then
(1) the h-line H for which iH interchanges P and Q is the h-bisector
of the h-segment PQ.
(2) the point where the h-bisector of PQ meets PQ is the h-midpoint of PQ.

The justification for the names h-bisector and h-midpoint will become clear
when we discuss hyperbolic distance.

These Lemmas can be used to prove fundamental results on
hyperbolic geometry. The Origin Lemma is often used to map
a chosen point to O to simplify the figure.

The Incidence Theorem

  1. two distinct points define a unique h-line,
  2. two distinct h-lines meet in at most one point.
To illustrate the use of the Lemma, we give the proof in full.

(1) Suppose that P and Q are distinct points of D.
By the Origin Lemma, there is a t in H(2) with t(P) = O.
Now, any h-line through O is a diameter of C.
There is a unique diameter H through t(Q).
Then t-1(H) is the required h-line.
(2) If two h-lines meet in two (or more) points, then, by (1),
they are identical. Thus, distinct h-lines meet at most once.

Note that these are exact analogues of incidence results in
euclidean geometry. However, the possibilities for non-intersecting
h-lines are rather different.
This has been discussed in the experimental pages.

There are many analogues of euclidean results. We give some
which will be useful later.

The Angle Bisector Theorem
If P, Q and R are points of D, with R not on ther h-line PQ,
then there is a unique h-line H bisecting <QPR.

This is left as an exercise to the reader. Simply apply the Origin Lemma
to map P to O and use the facts that h-lines through O are diameters,
and that hyperbolic transformations preserve the size of angles.

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.

Proof of the Perpendicular Theorem

In the figure, you can move the point P,
or vary the green h-line by moving A or B.
The red h-line is the perpendicular through P.

In euclidean geometry, we say that a figure P is
symmetric about a line L if reflection in L maps P to itself.

A figure P is symmetric about an h-line H if iH maps P to itself.

Symmetry of h-lines
An h-line L is symmetric in an h-line H if and only if L is perpendicular to H.

This is a consequence of the Mirror Property from inversive geometry.

We shall meet more interesting examples when we look at hyperbolic
circles and triangles.

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