Some of the difficulties of the different plane models for the hyperbolic
plane are readily apparent. The Klein model is not conformal--angles are not
preserved--we cannot use a Euclidean protractor--and it distorts distances.
The Poincaré disk model, while conformal, distorts distances--we cannot use
a Euclidean ruler. People have searched for another model which is conformal
and that represents hyperbolic lengths faithfully by Euclidean lengths. Such a
model would be called isometric . We are able to construct
a model of
elliptic geometry on the surface of the sphere in 3-space. Is there a surface
in 3-space on which we can isometrically model hyperbolic geometry? If so, the
lines of the hyperbolic plane would then be represented by
geodesics 
on the surface. Since
there seems to be some feeling that hyperbolic lines cannot be straight (
i.e., modelled on Euclidean lines), we might expect this surface to be curved
in Euclidean 3-space.
A theorem first proved by David Hilbert states that it is impossible to embed the entire hyperbolic plane isometrically as a surface in Euclidean space. On the other hand it is possible to embed the Euclidean plane isometrically in hyperbolic space, as the surface of the horosphere.
It is possible to embed a portion of the hyperbolic plane isometrically in Euclidean space, the portion called a horocyclic sector, bounded by an arc of a horocycle and the two diameters of the horocycle cutting off this arc. Figure 20.1 shows such a sector in the Poincaré model.
Figure 20.1: Horocyclic Sector in the Poincaré model
We must embed this sector isometrically onto a surface in Euclidean 3-space.
The surface that represents this region is called a
pseudosphere . It is
obtained by rotating a curved known as the tractrix around its
asymptote. The tractrix is the curve characterized by the condition that the
length of the segment of the tangent line to the curve from the curve to the
y-axis is constant. It has the following equation for a given constant a:
and has graph shown in Figure 20.2.
The representation on the pseudosphere gives us the opportunity to give some
geometric meaning to the fundamental constant k in the theorem on area.
Gauss discovered a way in which we can measure the
curvature of a
surface. Intuitively, it is a number K which measures how much a surface
bends. The sphere of radius R, for example, has constant curvature 
.
This makes sense as you think about, because the larger a sphere is the less
it bends. In general the curvature will change from point to point, or region
to region, on a surface. It should be zero at points where the surface is flat
and it should be large where the surface bends sharply. Surfaces where the
curvature is the same at all points are called surfaces of constant
curvature. The sphere is such an example.
For a surface of constant curvature K, Gauss found a fundamental formula
which relates the curvature, area, and angular measure. Take a geodesic
triangle, 
--a triangle where the sides are geodesic segments.
Let 
denote the radian measure of angle A. Gauss showed that
What does this mean? There are three possible cases.
Case 1. Let K be positive. This means that both sides of the
above equation are positive. This means that the angle sum in radians of a
geodesic triangle is greater than 
( or in degrees, greater than
), and the area of the triangle is proportional to the
excess.
On a sphere of radius R, then we know that the area of a geodesic triangle
is
If we have a geodesic triangle on the surface of the sphere with three right
angles, then the excess of this triangle is 
. Thus, the area of
this geodesic triangle is
A theorem by Liebmann, Hopf, and Rinow proves that spheres are the only
complete surfaces of constant positive curvature in Euclidean three-space. It
follows from this information that elliptic geometry cannot be embedded
isometrically in Euclidean three-space.
Case 2. K=0. Gauss' formula then says that the angle sum of a
geodesic triangle, in radians, is exactly . An example is the Euclidean
plane. Another example is an infinitely long cylinder.
Case 3. K<0. Now Gauss' formula shows that the angle sum in
radians is less than , and the area is proportional to the
defect.
The pseudosphere is a surface of constant negative curvature. Since the
pseudosphere represents a portion of the hyperbolic plane isometrically, we
can compare Gauss' formula with the formula for the area in the hyperbolic
plane. The comparison gives 
. Thus, 
is the curvature
of the hyperbolic plane.
Of interest, let r=ik where 
. Then 
, so that the
hyperbolic plane can be described as a sphere of imaginary radius
r=ik, as was noted by the mathematician Lambert. This description is very
reasonable, for all the formulas of hyperbolic trigonometry can be obtained
from the formulas known for spherical trigonometry by replacing r with ik.
Finally, notice that as k gets very large, the curvature K approaches zero, and the geometry of the surface resembles more and more the geometry of the Euclidean plane. It is in this sense that Euclidean geometry is a limiting case of hyperbolic geometry.
