# 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). 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}. 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 For z,w ε D, D(z,w) = |z-w|/|w*z-1|. 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). 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). 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.