In our discussion of the euclidean circle through three points in the disk
which do not lie on a hyperbolic line, we used the device of transforming
the figure so that one point maped to O. Since we simply wanted to find
when the euclidean circle crossed C, we did not need to consider how the
results vary if we choose different points, or if we use different hyperbolic
Suppose that K is a non-trivial intersection of an i-line with the disk D.
Choose a point A on K and hyperbolic transformations f,g which map
A to O. Then, j = gof-1 fixes O, so is also a euclidean transformation,
a rotation about O or a reflection in a diameter. Now g = jof, so
that g(K) is obtained by applying the euclidean j to f(K). It follows that
g(K) and f(K) have the same euclidean radius (or are lines) Thus, the
value obtained for the radius does not depend on the choice of map.
Now suppose that B is a second point on K. From the symmetry lemma,
there is a hyperbolic transformation h mapping B to A and K to K. Then
foh maps B to O, and maps K to foh(K) =
f(h(K)) = f(K). Thus the radius
of the euclidean circle does not depend on the choice of A.
In the proof of the circumcircle theorem, we computed the radius R of
a euclidean circle through three points, one of which was O. We obtained
2RΔ = 4s(AB)s(BC)s(CA), where s(PQ)=sinh(½d(P,Q)), and Δ2 can be
written as a function symmetric in s(AB), s(BC), s(CA). We will not look
at cases where A,B,C lie on a hyperbolic line (when Δ = 0). Then the
remarks above show that R will be the same for any three points on an
i-line (not orthogonal to C). It is actually better to use not R but rather
the function S = 1/4R2-1, which determines the positive number R.
After some algebraic simplification, we see that
S = J(A,B,C) = H(s(AB),s(BC),s(CA))/4s2(AB)s2(BC)s2(CA),
where H is the Heron Function H(x,y,z)=(x+y+z)(x+y-z)(x-y+z)(-x+y+z).
Notice that J is a hyperbolic invariant.
the j-invariant theorem
If A,B,C lie on K, the intersection of an i-line and D, then
(1) the value of J(A,B,C) is independent of the choice of A,B,C,
We shall write the common value as j(K).
(2) K is a hyperbolic circle if j(K) > 0, a horocycle if j(K) = 0,
and a hypercycle if j(K) < 0,
(3) K and L are hyperbolic congruent if and only if j(K) = j(L).
To work on hyperbolic polygons with vertices on an i-line, we need the idea
of "expanding" an i-line. The invariant j helps.
In our work on hyperbolic circles, we discussed the arcs determined by
two points. These were characterized by the value of the function E.
There are analogues for horocycles and hypercircles.