The hyperbolic lines were defined as the intersections of D with i-lines orthogonal to C.
It is reasonable to ask whether intersections with other i-lines are also significant in
Already we know that i-lines which are circles lying entirely within D are precisely the
We have also met i-lines which are circles touching C internally at a point X. These are
the horocycles at X. They arose as the boundaries of regions containing points C for
which ABC is a hyperbolic cyclic triangle.
In the course of our work on Ptolemy's Theorem, we met the remaining family. These
are the intersections with i-lines which cut C twice, but are not orthogonal to C. These
are the hypercircles. In the context of Ptolemy's Theorem, they simply arose as part
of the class of functions to which the theorem applied. They appear in a much more
The figure on the right shows a hyperbolic circle K, a horocycle E and hypercircles H
In euclidean and hyperbolic geometry, given a point P not on a line L, we can define
the hypercircle theorem|
Suppose that L is a hyperbolic line, with boundary points X and Y, and that P is a point
which does not lie on L. Then the locus C(P,L) is the hypercircle through X,Y and P.
Also, each hypercircle arises in this way for some P and L.
Although we shall not attempt to identify the symmetry groups of horocycles and
Note that, in the case of a hyperbolic circle, the hyperbolic perpendicular bisector
The lemma shows that the bisector of any chord will be a diameter in this sense
For a horocycle at X, the diameters obtained in this way all pass through X. In fact,
the symmetries of a horocycle.
The first part is easy - K and C meet only in X, and h also maps C to itself.
Our first application is to revisit the topic of hyperbolic circumcircles