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\pagestyle{fancy} \lhead{{\sf Homework 4 $|$ Due February 21 (Tuesday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 481 $|$ Topics in Geometry and Topology
$|$ Spring 2012 $|$ M.~Macauley}}
\begin{document}
$\;$
\emph{Read}: Stahl, Chapters 4.4., 5.1, 5.2, 5.3, 5.4.
\begin{enumerate}
% Stahl Ex 5.1.2 (p. 74).
\item Find the Euclidean center and radius of the circle that has
hyperbolic center $(5,4)$ and radius $3$.
% Stahl Ex 5.1.7 (p. 74).
\item Prove that the hyperbolic circumference of a circle with
hyperbolic radius $R$ is $2\pi\sinh(R)$.
\item Prove Proposition 5.1.4 in Stahl, that in hyperbolic geometry,
every angle is congruent to an angle in standard position. Recall
that in class we showed this for right angles, which was the
hyperbolic analog of Euclid's Postulate 4.
\item Let $P$ be a point in $\H^2$, and let $\ell$ be a Euclidean line
segment through $P$. Prove that there exists a unique hyperbolic
geodesic $g$ that passes through $P$ and is tangent to $\ell$. (In
other words, prove that, given a point $P$ and a unit vector $\v$,
there is a unique geodesic through $P$ that leaves $P$ in the
direction of $\v$.) Also, give an explicit construction/description
of $g$, given $\ell$, using either ruler-and-compass or analytic
methods.
% Stahl Ex 5.2.6. (p. 75)
\item Exercise 5.2.6 in Stahl asks you to consider the hyperbolic
analog of the Euclidean theorem which states that the tangent line
to a circle is perpendicular to the radius through the point of
contact. Specifically, determine the hyperbolic analogs of the
Euclidean terms in the statement of the Euclidean theorem (e.g.,
hyperbolic tangent line to a hyperbolic circle, radius of a
hyperbolic circle), and either prove that the analogouus statement
is true, or modifiy it as little as possible to make it true.
% Stahl Ex 5.2.7 (p. 75)
\item Suppose $g$ and $h$ are two geodesics in $\H^2$, which, as
figures of the Euclidean plane, are tangent to each other.
\begin{enumerate}
\item Prove that the point of tangency is on the $x$-axis.
\item Prove that the two geodesics are hyperbolically asymptotic
near that point of tangency.
\end{enumerate}
\smallskip
\emph{Problems $7$--$8$ outline proofs of the lemmas needed for the
``three reflections theorem'' of the hyperbolic plane.}
% Proof of Prop 3.3.1.
\item
\begin{enumerate}
\item Let $A$ and $B$ be distinct points in $\H^2$. Prove that the
geodesic segment from $A$ to $B$ has a unique (geodesic)
perpendicular bisector.
\item Let $A$ and $B$ be distinct points in $\H^2$. Prove that there
is a hyperbolic reflection $\rho$ such that $\rho(A)=B$.
\end{enumerate}
% Proof of Prop 3.3.2.
\item
\begin{enumerate}
\item Let $A$, $B$, and $C$ be distinct points in $\H^2$. Prove
that the (geodesic) angle bisector of $\angle BAC$ exists and is
unique.
\item Let $A$, $B$, and $C$ be distinct points in $\H^2$ such that
$h(A,B)=h(A,C)$. Prove that there is a hyperbolic reflection
$\rho$ such that $\rho(B)=C$ and $\rho(A)=A$.
\end{enumerate}
\end{enumerate}
\end{document}
% Future notes: Problems 4-8 should be on the same HW assignment (as 5
% and 8 reference 4)