the j-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).

proof
Most of the work has been done in the preliminary remarks.
(1) If we transform a point P on K to O, then the radius of the image
is fixed. All the points of K map to this image, so any three determine
its radius.
(2) The image is a hyperbolic circle if 2R < 1, a horocycle if 2R = 1
and a hypercircle if 2R > 1. These correspond to the given results.
(3) If K and L are congruent, then, since J is a hyperbolic invariant
and takes the value of j on each curve, j(K) = j(L).
On the other hand, if j(K) = j(L), then K and L can each be mapped
to a euclidean arc through O whose radius is determined by the
common value of j. These two arcs are clearly congruent, by a rotation
about O. It follows that K and L are congruent.

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