We have already indiated that there is a concept of distance in hyperbolic geometry, and that this is related to the function D.
Definition
In the hyperbolic circles pages, we introduced the function D.

For z, w in D_{0} D(z,w) = zw/w*z1

Definition For z, w in D_{0}, the hyperbolic distance between z and w given by d(z,w) = 2arctanh(D(z,w)).
Note that arctanh is an increasing bijection from [0,1) to [0,∞).
To verify that it satisfies D4, we begin by looking at the case
Lemma 1
Lemma 2 Together, these establish D4, so d is a distance function on the disk.

The hyperbolic radius of a hyperbolic circle.
The hcircle K(w,r) consists of points z with D(z,w) = r.
The CabriJava window shows an hsegment AB with D(A,B) = 1/2.
