# Hyperbolic Distance

 We have already indiated that there is a concept of distance in hyperbolic geometry, and that this is related to the function D. Definition If G is a group of transformations of a set S, then a distance function for the geometry on S is a function d from SxS to R such that,      for A, B, C in S, and g in G D1 d(A,B) ≥ 0, with equality if and only if A = B. D2 d(A,B) = d(B,A). D3 d(A,B) ≤ d(A,C) + d(C,B), with equality if and only if C lies between A and B. D4 d(g(A),g(B)) = d(A,B). In the hyperbolic circles pages, we introduced the function D. We observed that it satisfies D1, D2 and D4. A direct calculation shows that it does not satisfy D3. For z, w in D0 D(z,w) = |z-w|/|w*z-1| Definition For z, w in D0, 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,∞). It follows that d satisfies D1, D2 and D4. To verify that it satisfies D4, we begin by looking at the case where C lies between A and B. Lemma 1 If C lies between A and B, then d(A,B) = d(A,C) + d(C,B). Lemma 2 If C does not lie between A and B, then d(A,B) < d(A,C) + d(C,B). Together, these establish D4, so d is a distance function on the disk.
 The hyperbolic radius of a hyperbolic circle. The h-circle K(w,r) consists of points z with D(z,w) = r. In our new notation, it consists of the points z with d(z,w) = 2arctanh(r), so we say that K(w,r) has h-radius 2arctanh(r). Note that as r tends to 1, the h-radius tends to ∞, so we have arbirarily large hyperbolic distances in the disk. The CabriJava window shows an h-segment AB with D(A,B) = 1/2. Thus d(A,B) = 2arctanh(1/2) = ???. If you drag A, then B moves so that d(A,B) always has this value. You should observe that the representation appears smaller as it approaches the boundary.