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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)
