The work we have done so far allows us to describe the hyperbolic polygons with vertices on a horocycle. The result on euclidean circumcircles shows that A,B,C lies on a horocycle if and only if H(s(AB),s(AC),s(BC)) = 0, where s(PQ) = sinh(½d(P,Q)), and H(x,y,z) is the Heron polynomial. Thus one of s(AB),s(AC),s(BC) is the sum of the others. Observe that, if A,B,C lie on a horocycle K, then one of the points lies on the finite arc defined by the other two. The arcs theorem shows that, if C lies on the finite arc AB, then E(A,B,C) < 0, so s^{2}(AB) > s^{2}(AC)+s^{2}(CB). We must then have s(AB)=s(AC)+s(CB). This generalizes by induction on m to
the horocycle polygon theorem
Saying that the points lie in order means that, for each k, A(k) is on the finite arc
defined This has a valid converse :
the converse horocycle polygon theorem
This is proved by choosing A(0) on a horocycle K, and choosing A(i), i=1,..,m
arc length along a horocycle
As you might expect, we approximate L by n polygonal arcs each of length l_{n}.
