Models of the Hyperbolic PlaneRoss Hayter

Supervisor: Dr John Parker, Durham University

1. Introduction: No More Euclid!

Euclid’s fifth postulate states that two parallel lines will never intersect one another if the lines are extended to infinity. Many greatminds have looked at this statement and have been unable to prove it. In the first half of the nineteenth century mathematicansasked themselves: ‘What happens when Euclid’s postulate does not hold?’ At that moment non-Euclidean geometry was born and amathematical revolution had taken place.Hyperbolic geometry is a type of Non-Euclidean geometry that exists on negatively curved surfaces. One can tell if the surface isnegatively curved if the sum of the interior angles of a triangle are less than π.

2. The Hyperboloid Model

The first step of our hyperbolic journey isto define a space that allows negative cur-vature. In Lorentz 3-space a sphere withan imaginary radius r = i can be con-structed giving a Gaussian curvature of−1. In R3 this is equivalent to a Hyper-boloid defined by

z2 − (x2 + y2) = 1

where x, y, z are the standard cartesianco-ordinates. This surface separatesinto two sheets but it is only the posi-tive sheet z > 0 that defines the hyper-boloid model of the hyperbolic plane.

Figure 1: The positive sheet of the hyper-boloid (Maple).

When a Euclidean plane through the ori-gin intersects the hyperboloid a hyperbolais formed at this intersection. The hyper-bola formed defines a hyperbolic lines inthis model and is of the cartesian form

x2

a2− y2

b2= 1

where a and b are some constants. Thisis an ideal model of the hyperbolic planebut it is difficult to analyse as it is in threedimensional space.

Imagine looking up at the hyperboloid. Itwould appear as a circle even though thesheet heads towards infinity. A hyper-boloid and circular disc have similar topol-ogy so we can use stereographic projec-tion to create the next model.

3. Klein Disc Model

Let w = (x, y, z) where z2 > x2 + y2. Thenthere exists a unique point on the Eu-clidean line between (0, 0, 0) and (x, y, z)

that lies on the hyperboloid. If w is ac-tually on the hyperboloid and we con-sider the point (xz ,

yz, 1) which is situated on

a straight line through the origin and w.Substituting this point into the equation ofthe hyperboloid we get that

x2

z2+y2

z2+

1

z2= 1.

Therefore (xz ,yz) is on the interior

of the unit disc centred at (0, 0, 1).This is called the Klein Model.

Figure 2: Stereographic projection of thehyperboloid onto the Klein disc (xfig).

The hyperbolic lines in the hyperboloidmodel created by the intersection of theplanes ax + by + cz = 0 are mapped toaxz + byz + c = 0 which are open chords inthe unit disc known as K-lines.

4. Poincare Disc Model

We can create another model by pro-jecting the hyperboloid onto a discthat its centred at the origin fromthe projection point (0, 0,−1). Thisis called the Poincare Disc Model.

Figure 3: Stereographic projection of thehyperboloid onto the Poincare disc (xfig).

The lines in this model are arcs of circlesorthogonal to the circumference.

5. Limitations

Depending on the situation some modelsare more advantageous to use than oth-ers. The benefits and limitations of eachmodel are outlined in Figure 4.

Figure 4: Comparison of Models

Klein Model Poincare Disc ModelDistance distorted Distance distortedK-lines are P-lines are arcsEuclidean lines of circlesConvexity kept Convexity distortedNon-conformal Conformal

Different models are better when investi-gating properties of hyperbolic polygons.

Figure 5: Quadrilateral in the Klein Model(left) and the Poincare disc Model (right).

6. Escher: An Artist’s View

The concept of Hyperbolic space capturedthe imagination the artist Maurits CornelisEscher. This passion for Non-Euclideanart came directly from Poincare himself.Escher managed to translate complicatedmathematical ideas into remarkable piecesof art. M.C. Escher, Circle Limit III

7. The Universe

The discovery of hyperbolic geometry has made us question ourown universe. Is the universe Euclidean or Hyperbolic? Gaussattempted to answer this question by constructing a giant triangleusing the tops of three mountains as the vertices. Unfortunatelydue to experimental error his results were inconclusive. Do welive in a Euclidean world?

References: M. J Greenberg, Euclidean and Non-Euclidean Geometries, W.H. Freeman and Company,1980; J.G. Ratcliffe, Foundations of Hyperbolic Manifolds, Springer-Verlag, 1994