the hyperbolic plane

Here, we introduce the Poincare disc model of hyperbolic geometry.
Our description uses ideas from inversive geometry.

We can get a similar model
for euclidean geometry.
The set is the open disc D:{z : |z| < 1} in the complex plane.
Note. The boundary of D is the unit circle C: {z : |z| = 1}.
The points of C do not belong to the geometry, but they play
a role similar to the points at infinity in euclidean geometry.

Definition A hyperbolic line (or h-line) is a subset of D
of the form LnD, where L is an i-line orthogonal to C.

An i-line L is either an extended line or a circle. Observe that
an extended line L is orthogonal to C if and only if 0εL.
In such a case, the h-line LnD is a diameter of D.

The concept of orthogonal circles is less familiar. We have the following facts:

  1. If circle C (centre O) is orthogonal to circle L (centre P),
    then O lies outside L, and P lies outside C.
  2. If P is a point outside C, then there is a unique circle
    with centre P orthogonal to C.
proofs of these facts

The figure illustrates the h-line obtained from the point P.
The complete orthogonal circle is shown, h-line is shown in red.
You may move the point P to see how the h-line varies.
If you drag P inside C, there is no circle L, and hence no h-line.

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