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\begin{document}
\title{Modelling Hyperbolic Geometry}
\author{Robert D. Borgersen\\
umborger@cc.umanitoba.ca\\
Supervisor: Dr. William Kocay\\}
\date{February 14, 2005}
\maketitle
\begin{slide} \label{slide:one}
\heading{Abstract}
In Euclidean geometry we have the following: Given a line L and a
point P not on it, there is exactly one line through P that is
parallel to L. It was discovered that assuming this is false
produces the equally valid Hyperbolic Geometry, where there are in
fact infinitely many lines through P that are parallel to L. This
presentation is an introduction to Hyperbolic Geometry, and on
modelling it in the Euclidean plane.
\end{slide}
\begin{slide} \heading{Outline}
\begin{MyList}
\item Introduction to Hyperbolic Geometry
\begin{itemize}
\item Interesting Results
\end{itemize}
\item Metrics and Geodesics
\item Hyperbolic Isometries
\item Modelling Hyperbolic Geometry
\item Program Demonstration
\item Tessellations (Time Permitting)
\end{MyList}
\end{slide}
\begin{slide} \heading{Euclidean Geometry}
Euclidean geometry rests upon the following axioms:
\begin{MyList}
\item Each pair of points can be joined together by one and only
one straight-line segment.
\item Any straight-line segment can be indefinitely extended in
either direction.
\item There is exactly one circle of any given radius with any
given center.
\item All right angles are congruent to one another, and finally,
\item Given a line L and a point P not on it, there is exactly one
line through P that is does not intersect L.
\end{MyList}
The fifth is known as the Parallel Postulate. It has been shown
that this fifth axiom is independent of the others. So what if it
wasn't there?
\end{slide}
\begin{slide} \heading{Absolute Geometry}
The first four axioms by themselves are sufficient to prove many
classical theorems true in geometry. Some of these are:
\begin{MyList}
\item If two lines intersect, they intersect at only one point.
\item On the line L from A to B, there is exactly one point
equidistant from A and B, and this point is between A and B.
\item The Side-Angle-Side, Angle-Side-Angle, and Side-Side-Side
rules for triangles.
\item If one angle of a triangle is not acute then the other two
angles of the triangle are acute.
\item The sum of the lengths of each two sides of a triangle is
greater than the length of the third side (Triangle Inequality).
\end{MyList}
\end{slide}
\begin{slide} \heading{Absolute Geometry continued}
It is important to note that some theorems require the parallel
postulate to be true, and some do not. This is essentially where
Hyperbolic Geometry comes from. When the parallel postulate is
true, the five axioms give us the usual Euclidean geometry. When
false, they produce two different non-Euclidean geometries:
Elliptic (or Projective) and Hyperbolic.
Since Absolute Geometry is based only on the first four axioms,
all facts true in Absolute geometry are true in the Euclidean,
Hyperbolic, and Elliptic geometries.
\end{slide}
\begin{slide} \heading{The Hyperbolic Axiom}
In Hyperbolic Geometry, the Parallel Postulate is replaced with
the following axiom:
If point P is not on a line T, then there are at least 2 lines
through P that do not intersect T.
Using this axiom, we can prove many theorems that, in Euclidean
geometry, are absolute nonsense.
\subheading{The Elliptic Axiom} If point P is not on a line T,
then there are no lines through P that do not intersect T. (i.e.
all lines intersect)
\end{slide}
\begin{slide} \heading{Extension of the Hyperbolic Axiom}
In fact, it can be shown that when you assume that there are at
least 2 lines through P that do not intersect T, it can be proven
that in fact there are infinitely many lines through P that do not
intersect T.
We can also separate this class of lines into two groups: Hyper
Parallel and Disjointly Parallel lines. Hyper parallel lines are
those parallel lines that intersect only at infinity--the distance
between the lines goes to zero as you move towards infinity.
Disjointly parallel lines are all others.
For a list of interesting unintuitive theorems true in Hyperbolic
Geometry, see [Kelly]
\end{slide}
\begin{slide} \heading{Metrics and Geodesics}
A Geodesic is defined as the curve of shortest length between two
points. In Euclidean geometry (the Cartesian Model), Geodesics
are of the form $y=mx+b$, the classical definition of a straight
line. In order to talk about the definition, we need a well
defined distance function: a metric.
\end{slide}
\begin{slide}
\subheading{Metric} Formally, a Metric is a function d, defined on
a set, satisfying:
\begin{MyList}
\item $d(x, x) = 0$
\item $d(x, y) = d(y, x)$
\item $d(x, z) \leq d(x, y) + d(y, z)$
\end{MyList}
It is a distance function, describing the distance between any two
points in a set. In Euclidean Geometry, we have the following
metric: $$d(A,B) = \sqrt{(a_x-b_x)^2 + (a_y-b_y)^2}$$
\end{slide}
\begin{slide} \heading{Models of Hyperbolic Geometry}
It is one thing to develop Theorems in this geometry, but it's
another to perform constructions. A Euclidean model in which we
could use a compass and straight edge construction would be quite
helpful. Over the years, numerous models have been developed, but
I will present the most popular ones here.
\subheading{Poincare (interior to the) Disk Model} This is the
space $\{(x,y)|x^2+y^2<1\}$ (all the points within the unit
circle) together with the metric $$ds^2 =
4\left(\frac{dx^2+dy^2}{(1-x^2-y^2)^2}\right)$$
\end{slide}
\begin{slide}
In this model, the geodesics are the circular arcs orthogonal to
the unit circle, combined with the diameters of the circle (which
can be viewed as circular arcs with infinite radius). (Note that
this seems to be the most popular model for Hyperbolic
Geometry--see the artwork of Escher)
\subheading{Upper Half Plane Model} This is the space
$\{(x,y)|y>0\}$ (the upper half plane) together with the metric
$$ds^2 = \frac{dx^2+dy^2}{y^2}$$
\end{slide}
\begin{slide}
In this model, the geodesics are the upper half portions of
Euclidean circles of the form $(x-b)^2 + y^2 = r^2$ for some b and
some r. We also allow r to be infinite, and get Euclidean vertical
lines x = a for any a.
\subheading{Klein-Beltrami} This is the space
$\{(x,y)|x^2+y^2<1\}$ (all the points within the unit circle)
together with the metric
$$ds^2 = \frac{dx^2+dy^2}{1-x^2-y^2} + \frac{xdx^2+ydy^2}{(1-x^2-y^2)^2}$$
\end{slide}
\begin{slide}
In this model, the geodesics are the segments of Euclidean
straight lines that lie inside the unit circle.
\subheading{Logarithmic Half Plane Model} This is the space $R^2$
(the full plane) together with the metric
$$\frac{dx^2+e^{2y}dy^2}{e^y}$$
This model is directly related to the Half Plane model. In order
to help understand Hyperbolic Geometry, we extend the Half Plane
model to the full plane by taking the natural logarithm of each
point in the upper half plane. This produces geodesics of the
form $$ln \sqrt{r^2 - (x-x_0)^2}$$ where $x_0$ and $r$ are the
center and radius of a related half circle in the upper half plane
model.
\end{slide}
\begin{slide} \heading{Quote}
\begin{quote}
[The Upper Half Plane Model] is the hyperbolic plane, as much as
anything can be, but we call it a model of the hyperbolic plane
because any surface isometric to [The Upper Half Plane Model] is
equally entitled to the name.
\begin{flushright}John Stillwell
\end{flushright}
\end{quote}
\end{slide}
\begin{slide} \heading{Isometries}
An Isometry is a distance preserving mapping.
In the Euclidean plane, there are four kinds of isometries -
reflection, translation, rotation and glide reflection. These are
the classic "movements" of the Euclidean plane. Similarly, there
are four kinds of isometries in hyperbolic space: circle inversion
(hyperbolic reflection), hyperbolic isometry (hyperbolic
translations), the parabolic isometry (rotations at infinity) and
the elliptic isometry (hyperbolic rotation).
\end{slide}
\begin{slide} \heading{References}
\begin{MyList}
\item Beardon, Alan (1983): \emph{The Geometry of Discrete
Groups}. Springer-Verlag New York Inc.; ISBN: 0-387-90788-2
\item Gans, David (1973): \emph{An Introduction to Non-Euclidean
Geometry}. Academic Press, Inc.; ISBN: 0-12-274850-6
\item Kelly, Paul (1981): \emph{The Non-Euclidean Hyperbolic
Plane}. Springer-Verlag New York Inc.; ISBN: 0-387-90552-9
\item Levy, Silvio (1997): \emph{Flavors of Geometry}.
Mathematical Sciences Research Institute; ISBN: 0-521-62048-1
\item Robles, Colleen: "The Hyperbolic Geometry Exhibit"\\
http://www.geom.uiuc.edu/$\sim$crobles/hyperbolic/
\item Stillwell, John (1992): \emph{Geometry of Surfaces}.
Springer-Verlag New York Inc.; ISBN: 0-387-97743-0
\item Weisstein, Eric W. et al. "Geodesic." From MathWorld--A
Wolfram Web Resource. http://mathworld.wolfram.com/Geodesic.html
\end{MyList}
\end{slide}
\end{document}