An Introduction to Hyperbolic Geometry

Contents

1 Brief History of Hyperbolic Geometry 3

2 Models of Hyperbolic Geometry 4 2.1 Beltrami-Klein Model . . . . . . . . . . . . . . . . . . . 4

2.2 Poincaré Disk Model . . . . . . . . . . . . . . . . . . . . 4 2.3 Poincaré Half-Plane Model . . . . . . . . . . . . . . . . 5 2.4 Lorentz Model . . . . . . . . . . . . . . . . . . . . . . . 5

3 Introducing the Poincaré Half-Plane Model 5 3.1 Basics of the Half-Plane Model . . . . . . . . . . . . . . 5

3.2 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.3 Rigid Motions . . . . . . . . . . . . . . . . . . . . . . . 7

3.4 Möbius Transformations . . . . . . . . . . . . . . . . 10

4 Closing Remarks 11

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Introduction

We can describe hyperbolic geometry by taking the familiar first four postulates of Euclidean geometry and negating the fifth postulate[4 p359]. To understand what this means, we must first see what the fifth axiom is.

Euclid’s Fifth Postulate [1 p8] That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles [in sum], the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

This axiom is commonly named the Parallel Axiom[1][4] and is equivalent to another postulate, Playfair’s postulate.

Playfair’s Postulate[1 p28] Given a line m and a point P not on m, there is a unique line n that contains P and is parallel to m.Proof of equivalence to Euclid’s fifth postulate. This is not given in this essay, but can be seen in Stahl’s The Poincaré Half-Plane, pp28-31.

The equivalence of these two postulates makes the naming of Euclid’s fifth postulate clear. As a result of the fifth postulate, we have a uniqueness of parallel lines to a straight line through points not on that line. By negating Playfair’s postulate, we also negate Euclid’s fifth postulate. So, we can construct a new fifth postulate:

Fifth Postulate of Hyperbolic Geometry Given a line m and a point P not on m, there is more than one line n that contains P and is parallel to m.

Euclid’s first four postulates and this new postulate form the basis of hyperbolic geometry.

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1 Brief History of Hyperbolic Geometry[*]

Many mathematicians since the time of Euclid believed that Euclid’s fifth postulate was dependent on his first four postulates. As such, some of them attempted to prove the dependence of the fifth postulate by assuming the negation of the fifth postulate and deriving a contradiction from it. Most notable in these attempts was Saccheri in the 18th century. However, he failed to derive any contradiction from the negation of the fifth postulate and only started on the road to forming non-Euclidean geometry. More details on his attempts can be seen in Stillwell’s Mathematics and Its History pp461-462, or even Saccheri’s paper including his attempts Euclides ab omni naevo vindicantus (1733).

In this way, non-Euclidean geometry started to be formed. But it was not until the 1820s that there was a major publicised contribution towards hyperbolic geometry, when Lobachevsky and Bolyai independently published works on hyperbolic geometry. Their work unified many previous results on hyperbolic geometry and made much more thorough attempts to develop hyperbolic geometry than their predecessors. They discovered many of the results in hyperbolic geometry on triangles, circles and polygons. However, their work was largely overlooked or ignored until works on the subject by Gauss were released following his death.

With interest rekindled, Beltrami went on to form the first model of hyperbolic geometry, the projective model. From here, more models were developed and studied in great depth by various mathematicians. By the end of the 19th century, there had been a great leap in discoveries surrounding hyperbolic geometry.

_________________ * The majority of this section is taken or adapted closely from Stillwell’s Mathematics and Its History §18.1-18.5.

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2 Models of Hyperbolic Geometry

In this section I will briefly describe the various models that we use to visualise and understand hyperbolic geometry.

2.1 Beltrami-Klein Disk Model

This is a model of hyperbolic space based on the open unit disk Dn of Euclidean space of dimension n[2 p75]. It was first developed by Beltrami and was also the first model of hyperbolic geometry. Beltrami also realised that the rigid motions of this model were the projective transformations which map the unit disk to itself. Cayley had previously used these transformations to define a metric, the Cayley-Klein metric[4 p368].

Figure 2.1.1[4 p368]

Figure 2.1 shows two straight lines through a point P which are parallel to the straight line L, exhibiting the failure of the parallel axiom.

2.2 Poincaré Disk Model

This model again uses the open unit disk Dn of Euclidean space of dimension n. The distance attached to this model is given by:

cosh d (P ,Q )=1+2 ¿ P−Q∨¿2

(1−|P|2)(1−|Q|2)¿ where P,Q are points in the disk[2 p76].

In this model, the geodesics are just the diameter of the disk or arcs contained in the disk orthogonal to the boundary of the disk[2 p76]. The artist Escher illustrates this model in his prints Circle Limit III and Circle Limit IV.

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P

L

Figure 2.2.1: Escher’s Circle Limit III

2.3 Poincaré Half-Plane Model

The Poincaré half-plane modeluses the open upper half-plane, denoted H2, of C and the distance given by[2 p80]:

cosh d (z ,w )=1+ ¿w−z∨¿2

2ℑ [w ] ℑ[ z ]; z ,w ϵ H 2¿

This is the model I will be concentrating on in my essay, so I will leave out any further details.

2.4 Lorentz Model

This model in three dimensions uses the upper sheet of the hyperboloid of two sheets given byqL ( v )=−1, where qL ( v )=−t 2+x2+ y2 with t > 0.[5 p53]

So, H 2= {(t , x , y )|−t2+x2+ y2=−1∧t>0 }.

This space can be thought of as a hyperboloid of two sheets embedded in Minkowski space.

3 Introducing the Poincaré Half-Plane Model

3.1 Basics of the Half-Plane Model

As already said, the Poincaré half-plane model has its space being the upper half-plane of C, the complex space. We denote the space of the Poincaré half-plane H2. So, we can define our space.

Definition 3.1.1. The upper half plane H2 = { z ϵ C|ℑ ( z )>0 }

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Stahl suggests imagining that as you approach the real axis, space contracts, i.e. distance decreases. If we assume this contraction is roughly proportional to your distance from the real axis, we can conclude that it is impossible to reach the real axis.[1 pp63-64[ This is best illustrated by a series of lines equidistant apart.

If we take z = x + iy, this means that roughly

Hyperbolic Lengt h= EuclideanLengt hy

, with constant of proportionality 1.

This is not exactly correct, but gives a rough idea of what is going on. Now,

Euclidean length of an infinitesimal line segment is just √d x2+d y2, so with our

rough hyperbolic length, the length of such an infinitesimal line segment is

ds=√d x2+d y2

y. This is actually the metric used in this space.

Definition 3.1.2. The infinitesimal metric used in the Poincaré half-plane model

is ds=¿dz∨ ¿ℑ(z )

¿ [4 p375]

Using this infinitesimal metric, we can compute hyperbolic lengths of paths in the half-plane.

Definition 3.1.3. Hyperbolic length of an arbitrary curveγ : [ t0 , t1 ] →H2 is given

by[1 p65]

l (γ ) :=∫γ

√d x2+d y2

y=∫

t 0

t 11

y (t) √( dxdt

)2

+( dydt

)2

dt

3.2 Geodesics

We have defined the half-plane and have a concept of hyperbolic length, but in Euclidean geometry we have straight lines, or geodesics. So what are the geodesics of our Poincaré half-plane?

Definition 3.2.1. A “straight line” or geodesic between two points is a path joining the two points of minimal length[4 p346]

Theorem 3.2.1. The geodesic segments of the hyperbolic plane are either

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a) segments of Euclidean straight lines that are perpendicular to the real axisorb) arcs of Euclidean semicircles that are centred on the real axisProof.[1 p67] Let z1=x1+iy1 and z2=x2+iy2 be two points in our space H2. Now consider an arbitrary pathγ connecting z1 and z2. We have two cases, when x1= x2 and when x1≠ x2. Case 1: Assume that x1= x2. Using our definition of the hyperbolic length of a

curve, l (γ )=∫y1

y21y√(dx )2+(dy )2

¿∫y1

y21y √( dxdy )

2

(dy )2+(dy )2

¿∫y1

y21y √( dx

dy)2

+1dy

≥∫y1

y21ydy

This provides a lower bound for the length of the path, and we have equality

whendxdy

=0, i.e. γ is a vertical line. Therefore, we have found that in this case

the geodesic is a segment of a Euclidean straight line that is perpendicular to the real axis, so part a) is shown.

The proof that the second case results in the geodesic described in part b) of the theorem is similar, and can be seen in Stahl’s The Poincaré Half-Plane pp67-68.□

Definition 3.2.2. Hyperbolic distance between two points a,b in H2, denoted dH (a ,b), is defined to be the hyperbolic length of the geodesic joining a and b.

3.3 Rigid Motions

We already have a definition of rigid motions of the Euclidean plane:

Definition 3.3.1.[1 p36] A rigid motion of the Euclidean plane is a transformation f(P) of the plane into itself such that d(P,Q) = d(P’,Q’) whenever P’ = f(P) and Q’ = f(Q), where d(P,Q) is the distance between two points.

We can also show that all rigid motions of Euclidean space are generated by reflections and that they are categorised into reflections, translations, rotations and glides.[1 §2] So do we have a similar concept in our model?

In order to describe the rigid motions of hyperbolic geometry, if there are any, it is necessary to make a few definitions from Euclidean geometry.

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Definition 3.3.1. A reflection in a line m is is a rigid motion such that every point on m is fixed, and each point not on m is mapped to a unique point such that if P’=ρm ( P ) , then m is the perpendicular bisector of the line segment PP’..[1 p39]

Definition 3.3.2. Given a circle q with centre C and radius k, two points P and P’ are said to be symmetrical with respect to q if:

a) C, P, P’ are collinear with C outside the segment PP’and

b) d(C,P) . d(C,P’) = k2

Definition 3.3.3. An inversion, IC ,k , is a function such that if IC ,k ( P )=P ', then P and P’ are symmetrical with respect to the circle with centre C and radius k. [1

p52]

Definition 3.3.4. A translation, τ , is a rigid motion such that the line segments PP’ and QQ’ are of equal length and direction, where τ ( P )=P ' and τ (Q )=Q' .

We say τ=τ PP'=τQQ' .[1 p37]

With these definitions and using what we know about hyperbolic distance, we can understand the following theorem.

Theorem 3.3.1. The following transformations of the hyperbolic plane preserve hyperbolic lengths:a) inversions IC,K where C is on the real axisb) reflections ρm where m is perpendicular to the real axisc) translations τAB where AB is parallel to the real axisProof.[1 pp71-73] The proof can be seen in Stahl’s The Poincaré Half-Plane, and considers the mapping by these transformations of arbitrary curves. He uses the definition of hyperbolic length of a curve above to show that the length of the curve is equal to the length of the mappings of the curve for each case.□

Definition 3.3.4. A hyperbolic reflection is either a Euclidean reflection in a straight geodesic or an inversion centred at some point on the real axis.[1 p89]

If m is the straight geodesic meeting the real axis at real part M, then the reflection in m, denoted ρm, is defined to be just the Euclidean reflection in m.

If we consider the point z = x + iy in our plane, then to ρm(z ) = (M – x) + iy, which is equal to M - z. Assuming z is not on m, the bowed geodesic, p, through z that is centred at M, is orthogonal to m at their intersection, A. Then ρm ( p )=p and it follows that ρm(z ) is on p. Since ρm is a hyperbolic rigid motion, the hyperbolic distance between z and A must be equal to the

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hyperbolic distance between ρm(z ) and A. So, m perpendicularly bisects z and

ρm(z ), as with Euclidean reflections.

Now consider the inversion centred at a point C on the real axis. The inversion will fix every point in the bowed geodesic g which is a semi-circle centred at C. Say this semi-circle is of radius k. Fix a point z not on the fixed

bowed geodesic and say IC ,k ( z )=z ' . Then there exists a bowed geodesic q

between z and z’, which is necessarily orthogonal to g at the intersection of q and g, B. From Theorem 3.3.1, IC ,k with C being a point on the real axis, IC ,k is a hyperbolic rigid motion, and so the hyperbolic distance between z and B must equal the hyperbolic distance between z’ and B. Therefore, g perpendicularly bisects z and z’, and so the inversion matches the analogue of Euclidean reflections.

So we now have three different types rigid motions of hyperbolic geometry; hyperbolic reflections in straight geodesics, hyperbolic reflections corresponding to an inversion and hyperbolic translations. If we consider the composition of two distinct hyperbolic reflections through lines intersecting at a point, we arrive at a natural extension of Euclidean geometry with the composition of hyperbolic reflections being a hyperbolic rotation. Since the composition map of two rigid motions is itself a rigid motion, this means that the hyperbolic rotation is also a rigid motion. Stahl shows in The Poincaré Half-Plane that in fact all rigid motions of hyperbolic geometry are compositions of hyperbolic reflections using the fact the property of every rigid motion being a composition of rigid motions is a property of abstract geometry and hyperbolic geometry is an abstract geometry.

Having discovered some hyperbolic rigid motions, and being able to find all of them by composing reflections, we now seek an algebraic description of the rigid motions. The following theorem helps to do this:

Theorem 3.3.2. The rigid motions of the hyperbolic plane coincide with the complex functions of the forms:

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i) f ( z )=αz+ βγz+δ

or ii) f ( z )=α (−z )+ βγ (−z)+δ

where α ,β , γ , δ are real numbers and αδ−βγ>0.Proof.[1 pp141-142] Horizontal translations map z = x + iy to z’ = (x + r) + iy, with r being a real number. Hence, horizontal translations can be expressed as

f ( z )= 1 z+r0 z+1

.

We now consider the reflections in straight geodesics. This is just a Euclidean reflection in a line m, which is a vertical line meeting the real axis at

the point z = M, with M being real. Therefore, f ( z )=r−z=1(−z )+r0 z+1

, for

some r a real number.Finally, consider the reflection in a bowed geodesic. This is an inversion

centred at some point a on the real axis, which is of the form:

f ( z )= k2

z−a+a=a z+k2−a2

z−a=

−a(−z )+k2−a2

−(−z¿)−a¿ .

Moreover, (−a ) (−a ) — (−1)(k2−a2)=k2+a2−a2=k2>0 ,since k>0, as required.

Since rigid motions are composed by reflections, it remains to show that compositions of these three rigid motions are of one of the forms given in the theorem. This is just computation, and is not given in this essay.□

3.4 Möbius Transformations

Definition 3.4.1[1 p142] A Möbius transformation is a map f :C →C of the form

f ( z )=az+bcz+d

where a ,b , c , d∈C and ad−bc≠0.

We can extend the complex plane C to Ĉ := C U {∞}, which is the stereographic projection of the real line with the point ∞. If we consider our f in definition 3.4.1. as f :Ĉ →Ĉ , then we must define f for certain points in the extended

complex plane. In the case that c ≠ 0, we define f ( dc )=∞ and f ( ∞ )=ac

. If c=0,

we define f(∞) = ∞. The inverse function of f is easy to calculate and you get that:

f−1(z)= dz−b−cz+a

. The inverse function exists for all z in the extended complex

plane using similar definitions for the cases a ≠ 0 and a=0 as above.The functions given in Theorem 3.3.2. are just special cases of Möbius

transformations. Since we want the function to map only to the upper half of the complex plane, we restrict α ,β , γ , δ to being real numbers and αδ−βγ>0

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. If we associate to each function f ( z )=az+bcz+d

the matrix A=(a bc d), then for

the rigid motions of hyperbolic geometry we require that det(A)>0 and A∈R2,2

, 2x2 matrices with real valued entries. We can normalise any such A to give that det(A)=1, and so the rigid motions of hyperbolic geometry are just

4 Closing Remarks

*

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References

[1] Saul Stahl: The Poincaré Half-Plane: A Gateway to Modern Geometry, John & Bartlett, 1993

[2] Birger Iversen: Hyperbolic Geometry, Cambridge University Press, 1992

[3] Harold Scott MacDonald Coxeter: Non-Euclidean Geometry, Toronto University Press, Fourth Edition 8

[4] John Stillwell: Mathematics and Its History, Springer, Third Edition 1977

[5] Miles Reid and Balázs Svendrői, Geometry and Topology, Cambridge University Press, 2005

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