Supervisor: Dr John Parker, Durham University · Supervisor: Dr John Parker, Durham University 1....
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Models of the Hyperbolic Plane Ross 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 great minds have looked at this statement and have been unable to prove it. In the first half of the nineteenth century mathematicans asked themselves: ‘What happens when Euclid’s postulate does not hold?’ At that moment non-Euclidean geometry was born and a mathematical 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 is negatively 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 is to define a space that allows negative cur- vature. In Lorentz 3-space a sphere with an imaginary radius r = i can be con- structed giving a Gaussian curvature of -1. In R 3 this is equivalent to a Hyper- boloid defined by z 2 - (x 2 + y 2 )=1 where x, y, z are the standard cartesian co-ordinates. This surface separates into 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 hyperbola is formed at this intersection. The hyper- bola formed defines a hyperbolic lines in this model and is of the cartesian form x 2 a 2 - y 2 b 2 =1 where a and b are some constants. This is an ideal model of the hyperbolic plane but it is difficult to analyse as it is in three dimensional space. Imagine looking up at the hyperboloid. It would appear as a circle even though the sheet 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 z 2 >x 2 + y 2 . Then there 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 ( x z , y z , 1) which is situated on a straight line through the origin and w . Substituting this point into the equation of the hyperboloid we get that x 2 z 2 + y 2 z 2 + 1 z 2 =1. Therefore ( x z , y z ) is on the interior of the unit disc centred at (0, 0, 1). This is called the Klein Model. Figure 2: Stereographic projection of the hyperboloid onto the Klein disc (xfig). The hyperbolic lines in the hyperboloid model created by the intersection of the planes ax + by + cz =0 are mapped to a x z + b y z + c =0 which are open chords in the unit disc known as K-lines. 4. Poincar´ e Disc Model We can create another model by pro- jecting the hyperboloid onto a disc that its centred at the origin from the projection point (0, 0, -1). This is called the Poincar´ e Disc Model. Figure 3: Stereographic projection of the hyperboloid onto the Poincar´ e disc (xfig). The lines in this model are arcs of circles orthogonal to the circumference. 5. Limitations Depending on the situation some models are more advantageous to use than oth- ers. The benefits and limitations of each model are outlined in Figure 4. Figure 4: Comparison of Models Klein Model Poincar´ e Disc Model Distance distorted Distance distorted K-lines are P-lines are arcs Euclidean lines of circles Convexity kept Convexity distorted Non-conformal Conformal Different models are better when investi- gating properties of hyperbolic polygons. Figure 5: Quadrilateral in the Klein Model (left) and the Poincar´ e disc Model (right). 6. Escher: An Artist’s View The concept of Hyperbolic space captured the imagination the artist Maurits Cornelis Escher. This passion for Non-Euclidean art came directly from Poincare himself. Escher managed to translate complicated mathematical ideas into remarkable pieces of art. M.C. Escher, Circle Limit III 7. The Universe The discovery of hyperbolic geometry has made us question our own universe. Is the universe Euclidean or Hyperbolic? Gauss attempted to answer this question by constructing a giant triangle using the tops of three mountains as the vertices. Unfortunately due to experimental error his results were inconclusive. Do we live 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
Transcript of Supervisor: Dr John Parker, Durham University · Supervisor: Dr John Parker, Durham University 1....
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
