Properties of hyperbolic circles

We have defined the h-circle K(γ,r) as the locus {z : D(z,γ) = r},
and have seen that an h-circle is a circle lying within D. However, it is
not clear that the h-centre and h-measure of an h-circle are well-defined,
i.e. that, if K(γ,r) = K(δ,s), then γ = δ and r = s.
It is not even clear that the h-centre lies inside the circle!

Indeed, if we define a euclidean circle as a locus {z : |z -γ| = r}, then
the problems arise even in euclidean geometry.

The nicest way to resolve these problems is to use the theory of apollonian families.

In the proof of the Lemma on h-circles, we observed that K(γ,r) belongs to the
apollonian family A(γ,1/γ*) if γ ≠ 0, and to A(0,∞) otherwise. Note that C
belongs to the family in either case. By the Common Inverses Theorem from
inversive geometry, two disjoint circles belong to a unique apollonian family,
so γ is uniquely defined by the locus K(γ,r). It follows easily that the radius
is unique since it is determined by the h-centre and any point on K(γ,r).

To proceed, we need some results from euclidean geometry:

Basic facts about euclidean circles

  1. any three non-collinear points lie on a unique circle,
  2. a line and circle meet in at most two points,
  3. two distinct circles meet in at most two points.
Since an h-circle is a circle, and an h-line is either a segment of a line,
or an arc of a circle, we have the

Basic facts about h-circles

  1. an h-line and an h-circle meet in at most two points,
  2. two distinct h-circles meet in at most two points,
  3. if an h-line contains an interior point P of an h-circle,
    then the h-line and h-circle meet in two distinct points,
    with P between these points.
Parts (1) and (2) are immediate from euclidean facts (2) and (3).
For (3), we need to observe that an h-circle K lies within D, so that any
h-line H must contain points outside K (nearer the boundary C). Thus,
if H contains an interior point P, then it meets K once on each side of P.

Fact (3) characterizes the interior of an h-circle in hyperbolic terms:
a point P lies in the interior of an h-circle K if and only if an h-line
through P cuts K at points Q and R, with P between Q and R.
It follows that any hyperbolic transformation t maps the interior of an
h-circle K to the interior of t(K).

In euclidean geometry, the centre plays a vital role in dicussion of
the symmetries of the circle, and of tangents.

Theorem 4
An h-circle K(P,r) is symmetric about the h-line H if and only if P lies on H.

Let h denote h-inversion in H.
By the Hyperbolic Circle Theorem, h(K(P,r)) = K(h(P),r).
Thus, we have symmetry about H if and only if K(P,r) = K(h(P),r).
Since the h-centre is unique, this occurs if and only if h(P) = P,
i.e. P is on H.

Basic Fact (2) leaves the possibility that an h-line and h-circle meet once.
As in euclidean geometry, we make the

If the h-line H meets the h-circle K in a single point P,
then H is a hyperbolic tangent (h-tangent) to K at P.

The Hyperbolic Tangents Theorem
If P lies on the h-circle K = K(Q,r), then
(1) there is a unique h-tangent to K at P, and
(2) the h-tangent is perpendicular to the h-line PQ.

By the Origin Lemma, there is an h-inversion h mapping P to 0.
This maps K to the h-circle K(h(Q),r) passing through 0.
Since any h-line through 0 is a diameter, there is a unique h-line
tangent to K(h(Q),r). Applying h-1, we get a unique h-tangent J
at P.
Let hQ denote h-inversion in the h-line PQ. Then hQ(K) = K
and hQ(P) = P. SInce the h-tangent is unique, we must have hQ(J) = J.
Thus J is perpendicular to PQ (see symmetry of h-lines).

return to hyperbolic circles