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.
Although circles map to circles, centres do not in general map to centres,
You can get a feel for this by trying experiments with circles
To investigate further, we need to know more about the elements of
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
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
This has the important corollary that the function D is invariant under H(2),
Properties of D
We shall pursue this further in the hyperbolic distance page.
main hyperbolic page