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
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
Lemma

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

This result suggests the following Definitions


From Theorem 3, we know that, as a set of points, an hcircle is a euclidean circle lying within D. Also, as D(z,0) = z0/0*z1 = z, the hcircle K(0,r) is just the euclidean circle C_{r} : z = r. By the Lemma, tεH(2) maps K(0,r) to K(t(0),r). In fact, elements of H(2) map hcircles to hcircles, and hcentre to hcentres.
The Hyperbolic Circle Theorem
This has the important corollary that the function D is invariant under H(2),
Corollary Proof of the Theorem and Corollary
Properties of D
We shall pursue this further in the hyperbolic distance page.
