7.2. MODELS OF HYPERBOLIC GEOMETRY
7.2.2
271
Mini-Project - The Klein Model
The Poincaré model preserves the Euclidean notion of angle, but at the
expense of defining lines in a fairly strange manner. Is there a model of
hyperbolic geometry, built within Euclidean geometry, that preserves both
the Euclidean definition of lines and the Euclidean notion of angle? Unfortunately, this is impossible. If we had such a model, and ∆ABC was any
triangle, then the angle sum of the triangle would be 180 degrees, which is a
property that is equivalent to the parallel postulate of Euclidean geometry
[Exercise 2.1.8 in Chapter 2].
A natural question to ask is whether it is possible to find a model of
hyperbolic geometry, built within Euclidean geometry, that preserves just
the Euclidean notion of lines.
In this project we will investigate a model first put forward by Felix
Klein, where hyperbolic lines are segments of Euclidean lines. Klein’s model
starts out with the same set of points we used for the Poincaré model, the
set of points inside the unit disk.
However, lines will be defined differently. A hyperbolic line (or Klein
line) in this model will be any chord of the boundary circle (minus its points
on the boundary circle).
Here is a collection of Klein lines.
Exercise 7.2.1. Show that Euclid’s first two postulates are satisfied in this model.
In order to verify Euclid’s third postulate, we will need to define a distance function.
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CHAPTER 7. NON-EUCLIDEAN GEOMETRY
Definition 7.4. The hyperbolic distance from P to Q in the Klein model is
dK (P, Q) =
¯
¯
1 ¯¯ (P S) (QR) ¯¯
ln(
)
2 ¯ (P R) (QS) ¯
(7.2)
where R and S are the points where the hyperbolic line (chord of the circle)
through P and Q meets the boundary circle.
Note the similarity of this definition to the definition of distance in the
Poincaré model. We will show at the end of this chapter that the Klein and
Poincaré models are isomorphic. That is, there is a one-to-one map between
the models that preserves lines and angles and also preserves the distance
functions.
Just as we did in the Poincaré model, we now define a circle as the set
of points a given (hyperbolic) distance from a center point.
Exercise 7.2.2. Show that Euclid’s third postulate is satisifed with this definition
of circles. [Hint: Use the continuity of the logarithm function, as well as the fact
that the logarithm is an unbounded function.]
Euclid’s fourth postulate deals with right angles. Let’s skip this postulate
for now and consider the hyperbolic postulate. It is clear that given a line
and a point not on the line, there are many parallels (non-intersecting lines)
to the given line through the point. Draw some pictures on a piece of paper
to convince yourself of this fact.
Now, let’s return to the question of angles and, in particular, right angles.
What we need is a notion of perpendicularity of lines meeting at a point.
Let’s start with the simplest case, where one of the lines, say l, is a diameter
of the Klein disk. Suppose we define another line m to be (hyperbolically)
perpendicular to l at a point P if it is perpendicular to l in the Euclidean
sense.
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7.2. MODELS OF HYPERBOLIC GEOMETRY
Shown here are several Klein lines
perpendicular to the Klein line l,
which is a diameter of the boundary circle.
O
l
It is clear that we cannot extend this definition to non-diameter chords
directly. If we did so, then right angles would have the same meaning in
the Klein model as they do in the Euclidean plane, which would mean that
parallels would have to satisfy the Euclidean parallel postulate.
The best we can hope for is an extension of some property of perpendiculars to a diameter. If we consider the extended plane, with the point at
infinity attached, then all the perpendicular lines to the diameter l in the
figure above meet at the point at infinity. The point at infinity is the inverse
point to the origin O, with respect to the unit circle (as was discussed at the
end of Chapter 2). Also, O has a unique position on l—it is the Euclidean
midpoint of the chord defining l.
If we move l to a new position, say to line l 0 , so that l0 is no longer a
diameter, then it makes sense that the perpendiculars to l would also move
to new perpendiculars to l 0 , but in such a way that they still intersected at
the inverse point of the midpoint of the chord for l 0 . This inverse point is
called the pole of the chord.
Definition 7.5. The pole of chord AB in a circle c is the inverse point of
the midpoint of AB with respect to the circle.
From our work in Chapter 2, we know that the pole of chord AB is also
the intersection of the tangents at A and B to the circle.
274
CHAPTER 7. NON-EUCLIDEAN GEOMETRY
Here we see a Klein line with Euclidean midpoint M and tangents
at A and B (which are not actually
points in the Klein model) meeting
at pole P . Notice that the three
chords (m1, m2, and m3) inside
the circle have the property that,
when extended, they pass through
the pole. It makes sense to extend
our definition of perpendicularity to
state that these three chords will
be perpendicular (in the hyperbolic
sense) to the given line (AB) at the
points of intersection.
B
m1
m2
O
M
P
m3
A
Definition 7.6. A line m is perpendicular to a line l (in the Klein model)
if the Euclidean line for m passes through the pole P of l.
Exercise 7.2.3. Use this definition of perpendicularity to sketch some perpendicular lines in the Klein model. Then, use this definition to show that the common
perpendicular to two parallel Klein lines exists in most cases. That is, show that
there is a Klein line that meets two given parallel Klein lines at right angles, except
in one special case of parallels. Describe this special case.
Given a Klein line l (defined by
chord AB) and a point P not on l,
there are two chords BC and AD,
both passing through P and parallel to l (only intersections are on
the boundary). These two parallels
possess the interesting property of
dividing the set of all lines through
P into two subsets: those that intersect l and those that are parallel to
l. These special parallels (AD and
BC) will be called limiting parallels
to l at P . (A precise definition of
this property will come later.)
C
P
D
B
l
A
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7.3. BASIC RESULTS IN HYPERBOLIC GEOMETRY
From P drop a perpendicular to
l at Q as shown. Consider the hyperbolic angle ∠QP T , where T is a
point on the hyperbolic ray from P
to B. This angle will be called the
angle of parallelism for l at P .
C
D
P
T
Q
B
l
A
Pole(AB)
Exercise 7.2.4. Considering the figure above, explain why the angle made by
QP T (the angle of parallelism) cannot be a right angle. Then use this result, and
the fact that AD and BC are limiting parallels, to show that the angle of parallelism
cannot be greater than a right angle.
7.3
Basic Results in Hyperbolic Geometry
We will now look at some basic results concerning lines, triangles, circles,
and the like, that hold in all models of hyperbolic geometry.
Just as we use diagrams to aid in understanding the proofs of results in
Euclidean geometry, we will use the Poincaré model (or the Klein model)
to draw diagrams to aid in understanding hyperbolic geometry. However,
we must be careful. Too much reliance on figures and diagrams can lead to
hidden assumptions. We must be careful to argue solely from the postulates
or from theorems based on the postulates.
One aid in our study of hyperbolic geometry will be the fact that all
results in Euclidean geometry that do not depend on the parallel postulate (those of neutral geometry) can be assumed in hyperbolic geometry
immediately. For example, we can assume that results about congruence of
triangles, such as SAS, will hold in hyperbolic geometry. In fact, we can
assume the first 28 propositions in Book I of Euclid (see Appendix A).
Also, we can assume results on isometries, including reflections and rotations, found in sections 5.1, 5.2, and 5.4, as they do not depend on the
parallel postulate. These results will hold in hyperbolic geometry, assuming
that we have a distance function that is well defined. We also assume basic