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\centerline{MONT 108N -- Mathematics Through Time}
\centerline{Discussion 5 -- A Euclidean Model of Hyperbolic Geometry}
\centerline{December 2, 2010}
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\noindent
{\it Background}
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The final step of showing that hyperbolic geometry is as consistent
as Euclidean geometry was provided by a number of different 19th century
mathematicians who showed that \emph{models} of hyperbolic geometry
could be constructed within the Euclidean plane.  If there were any 
contradictions, there would be contradictions in Euclidean geometry
as well. (This sort of argument is known as a {\it relative consistency
proof} in ``the foundations of mathematics biz.'')  Today, we will look
at one such proof that was provided by a 19th and 20th century
German mathematician named Felix Klein (1849 - 1925).  In the \emph{Klein
model} of hyperbolic geometry:
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\begin{itemize}
\item  A \emph{point} is a point of the disk $\mathcal{K}$ 
\emph{strictly inside} the circle of radius 1 with
center at $(0,0)$ in the coordinate plane.
\item A \emph{line} is the portion of a straight line in the plane
that lies inside $\mathcal{K}$. (Note:  points on the boundary circle $\mathcal{C}$
of the disk $\mathcal{K}$ are \emph{not part of the Klein model.})
\end{itemize}
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\noindent
{\it Discussion Questions}
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\begin{enumerate}
\item[A.]  Show that in this model, two \emph{points} determine one and 
only one \emph{line} (that is, a statement corresponding to Euclid's Postulate
1 holds).
\item[B.]  Show that in this model, given a \emph{line} $\ell$ and a \emph{point}
$P$, there are infinitely many different \emph{lines} $m$ containing $P$
that do not meet $\ell$.  (These are the parallels, so the hyperbolic
postulate (H) holds in this geometry.)
\item[C.]  To measure distances in $\mathcal{K}$, we proceed as follows:
Let $P,Q$ be \emph{points}, and using the Euclidean construction, extend
the (Euclidean) line $PQ$ until it meets the boundary circle in points 
$S,T$, in the order $S,P,Q,T$ along the (Euclidean) line.  Define:
\[
distance\ PQ = \ln\left(\frac{(QS)(PT)}{(PS)(QT)}\right),
\]
where the $(QS), (PT), (PS), (QT)$ are the Euclidean distances.     
\begin{enumerate}
\item[1.]  Show that if $P,Q,R$ are three points on a \emph{line}
in $\mathcal{K}$, then 
\[
distance\ PQ + distance\ QR = distance\ PR.
\]
\item[2.]  Let $P$ be fixed and let $Q$ move along the \emph{line}
toward the point $T$ on the boundary circle as in the definition
of the distance.  Show that $distance\ PQ \to \infty$.
Hence something like Euclid's Postulate 2 holds here as well.
\item[3.]  Finally, explain how you would define and construct
a \emph{circle} in this geometry (Postulate 3).
\end{enumerate}
In case you are curious here, to define angles in this model, we
must proceed by a \emph{different method} from the usual Euclidean
one.  This is a bit too technical for us to get into.  However,
with the correct definition, it can be seen that all right angles
are equal, so the analog of Postulate 4 holds as well!
\end{enumerate}
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