We begin by looking at cyclic polygons. The results for triangles suggest that the
results for hyperbolic lengths l(1),..,l(n) are related to results for euclidean lengths
s(l(1)),..,s(l(n)), where s(l) = sinh(½l).
For quadrilaterals, there is an analogue of ptolemy's theorem, again involving
the values s(l). There is even a hyperbolic extension of fuhrmann's theorem.
Using inversive geometry, we discover that ptolemy's theorem is actually valid
for i-linear polygons. Indeed, the i-linear polygons a re precisely those for which
the analogue of ptolemy holds.
Returning to the cyclic case, we show that there are analogues of the familiar
euclidean results about angles on an arc. These involve a function E which, in
the euclidean setting, can be interpreted as the cosine of an angle.
We show that there is a cyclic polygon with sides of hyperbolic lengths l(1),..,l(n)
if and only if there is a cyclic polygon with sides of euclidean lengths s(l(1)),..,s(l(n)).
We offer two proofs. One is the analogue of a euclidean proof, mainly algebraic.
It works only for quadrilaterals, and is related to ptolemy's theorem.
The other involves a strange mapping from part of the hyperbolic plane to part
of the euclidean plane. This mapping takes circles to circles. The second proof is
more powerful. It gives the result for arbitrary polygons.
We now discuss convexity in the hyperbolic plane. The theory is really the same
as that in the euclidean plane. The one extension we want is to include regions
bounded by "segments" of i-lines. This leads us into the concept of hypercircles
and horocycles. These objects appear in our final result, so we pursue the theory
to some extent. Parts of this discussion are side-roads. We shall map them later.
Returning to convexity, we present a geometrical approach, illustrated with CabriJava.
This consists of expanding a circle with successive chords of lengths l(1),..,l(n) and
looking at the distance between the end points of this polygonal arc.
In euclidean geometry, this can be turned into a formal proof. The difficult part is
the proof that the result is unique in the second of the cases.
In hyperbolic geometry, the euclidean circle used may not be a hyperbolic circle.
it may touch the boundary (a horocycle) or cut it twice (a hypercircle). But the
existence of a suitable arc is quickly established. The uniqueness requires more
of the theory of horocycles and hypercircles. This is eventually achieved by
a monotonicity theorem, showing that, in the difficult case, the distance between
the end points of a polygonal arc increases. There are three arguments. The first
applies while the euclidean circle is a hyperbolic circle. The third applies when it is
a hypercircle. The second shows that it is continuous at the transition (a horocycle).
We are then in a position to prove the major result :
The hyperbolic polygon theorem
Suppose that n ≥ 3 and that l(1),..,l(n) are positive real numbers.
- There exists an (l(1),..,l(n)) hyperbolic polygon if and only if B(l(1),..,l(n)) > 0,
- If B(l(1),..,l(n)) > 0, then there is a convex (l(1),..,l(n)) hyperbolic polygon,
- If B(l(1),..,l(n)) > 0, then there is an (l(1),..,l(n)) hyperbolic polygon whose
vertices lie on an i-line K. Moreover,
- K is unique up to hyperbolic conjugacy,
- K is a hyperbolic circle if B(s(l(1)),..,s(l(n))) > 0,
- K is a horocycle if B(s(l(1)),..,s(l(n))) = 0,
- K is a hypercircle if B(s(l(1)),..,s(l(n))) < 0.