the function E and arcs in general

 The function E characterises the arcs of a hyperbolic circle. In fact, it also gives resluts for horocycles and hypercircles. Recall that F(x,y,z) = (x2+y2-z2)/2xy, that 1 - F2(x,y,z) = H(x,y,z)/4x2y2, and that for A,B,C in the disk, E(A,B,C) = F(s(AC),s(BC),s(AB)). Also, J(A,B,C) = H(s(AB),s(BC),s(CA))/4s2(AB)s2(BC)s2(CA). Suppose that K is a horocycle or hypercircle and that A,B lie on K. The i-line defining K will meet C. It follows that A and B divide K into three arcs. The euclidean arc AB of the i-line which lies inside C, and the two arcs which meet the boundary. The former is the finite arc AB. The latter are the infinite arcs. Note that the infinite arcs contain points arbitrarily close to C, i.e. have points D with d(A,D) and d(B,D) arbitrarily large. The figure shows a hypercycle K through A and B in the disk. The i-line determining K meets the boundary at X and Y. The finite arc AB is coloured blue. The infinite arc AX (red) is the arc beyond A. The infinite arc BY (green) is the arc beyond B. the arcs theorem Suppose that K is a horocycle or hypercircle, and that A,B are points on K. Then, for any C on K other than A and B, (1) E2(A,B,C) is independent of C, and (2) E(A,B,C) < 0 if and only if C lies on the finite arc AB. This leads to useful results on horocycles.