Hyperbolic circles

In euclidean geometry, we define circles as loci in terms of distance.
Here, it is more convenient to begin with the loci, and using these to
develop the theory of hyperbolic distance.

Theorem 3
If K is a circle lying within D, and tεH(2), then
(1) t(K) is a circle lying within D, and
(2) the interior of K maps to the interior of t(K).

Proof of Theorem 3

Although circles map to circles, centres do not in general map to centres,
and the radius may change.

You can get a feel for this by trying experiments with circles

To investigate further, we need to know more about the elements of
the hyperbolic group

Lemma
For 0 < r < 1, tεH(2) maps the circle Cr = {z : |z| =r}
to the locus {z : |z - t(0)|/|t(0)*z - 1| = r}.

Proof of the lemma

Note that if γ = t(0) ≠ 0,
then the equation may be written
|z - γ| = (r|γ|)|z - 1/γ*).
Hence, the locus belongs to the
apollonian family A(γ,1/γ*).
It is a circle as r|γ| < 1.

This result suggests the following

Definitions

  1. For z,w ε D, D(z,w) = |z-w|/|w*z-1|.
  2. For z ε D and 0 < r < 1,
    the h-circle with h-centre γ and h-measure r
    is the locus K(γ,r) = {z : D(z,γ) = r}.
From Theorem 3, we know that, as a set of points, an h-circle
is a euclidean circle lying within D.
Also, as D(z,0) = |z-0|/|0*z-1| = |z|, the h-circle K(0,r) is just
the euclidean circle Cr : |z| = r.
By the Lemma, tεH(2) maps K(0,r) to K(t(0),r). In fact,
elements of H(2) map h-circles to h-circles, and h-centre to h-centres.

The Hyperbolic Circle Theorem
If γεD, 0 < r < 1, and tεH(2), then t(K(γ,r)) = K(t(γ),r).

This has the important corollary that the function D is invariant under H(2),
i.e. defines an h-property.

Corollary
If z, w ε D and tεH(2), then D(t(z),t(w)) = D(z,w).

Proof of the Theorem and Corollary

Properties of D
For z,w ε D,

  • D(0,w) = |w|,
  • D(z,w) ≥ 0,
  • D(z,w) = 0 if and only if z = w,
  • D(z,w) = D(w,z).

Proof of the properties

We shall pursue this further in the hyperbolic distance page.
Our present knowledge allows us to proceed with the theory of hyperbolic distance,
but there are many interesting properties of hyperbolic circles.
Most are analogues of euclidean theorems.

main hyperbolic page