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\pagestyle{fancy} \lhead{{\sf Homework 3 $|$ Due February 7 (Tuesday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 481 $|$ Topics in Geometry and Topology
    $|$ Spring 2012 $|$ M.~Macauley}}


\begin{document}

$\;$

\emph{Read}: Stahl, Chapters 4.1, 4.2, 4.3.

\begin{enumerate}
  
\item In this problem, we will prove Proposition 3.1.3 in Stahl in a
  more rigorous and elegant fashion, using only the analytic methods
  we developed in class.
  \begin{enumerate}
  \item Let $f$ be a Euclidean isometry, let $p_0$ be a circle, and
    let $f(p_0)=p_1$. Prove that if $I_{C,k}$ maps $p_0$ to a circle,
    then $f\circ I_{C,k}\circ f^{-1}$ maps $p_1$ to a circle.
  \item Let $f$ be a Euclidean isometry, let $\ell_0$ be a line, and
    let $f(\ell_0)=\ell_1$. Prove the if $I_{C,k}$ maps $\ell_0$
    to a line, then $f\circ I_{C,k}\circ f^{-1}$ maps $\ell_1$ to a
    line. 
  \item Prove that if $p$ is a circle and $q$ is either a circle or a
    line, then there exists $g\in\Isom(\E^2)$ such that $g(p)$ is
    centered at the origin, and $g(q)$ is orthogonal to the
    $x$-axis.
  \item Prove that if $I_{O,k}$ maps circles and lines orthogonal
    to the $x$-axis to circles and lines, then for any $C\in\E^2$,
    $I_{C,k}$ maps circles and lines to circles and lines. Draw the
    appropriate commutative diagram illustrating your argument.
  \item Let $p$ be a circle. Prove the following:
    \begin{enumerate}
    \item If $p$ does not contain $C$, then $I_{C,k}(p)$ is a circle
      not containing $p$;
    \item If $p$ contains $C$, then $I_{C,k}(p)$ is a straight line
      not containing $p$.
    \end{enumerate}
  \end{enumerate}
  
  
\item In this problem we will prove Proposition 3.1.7 analytically:
  Let $p$ be the circle of center $C$ and radius $k$, and let $q$ be
  any other circle in $\E^2\setminus C$. Then $I_{C,k}$ sends $q$ to
  itself if and only if $q$ is orthogonal to $p$.
  \begin{enumerate}
    \item Prove that it is enough to prove that proposition for the
      case where the cetners of $p$ and $q$ are on the $x$-axis.
    \item Prove that the proposition for the case where the centers of
      $p$ and $q$ are on the $x$-axis. You may find the following
      facts useful:
      \begin{enumerate}
      \item Let $q$ be a circle with center $C$, and let $P$ be a
        point on $q$. Then the tangent line to $q$ at $P$ is
        orthogonal to $\overline{CP}$.
      \item In the triangle below, $a^2+b^2=c^2$ if and only if
        $\theta=\pi/2$. (This follows from the law of cosines.)
        \[
        \begin{tikzpicture}[scale=0.75]
          \draw {(0.1,0.4) node{$\theta$}};
          \draw {(-1.75,1.25) node{$a$}};
          \draw {(1.6,0.2) node{$b$}};
          \draw {(0.25,2.3) node{$c$}};
          \draw {[-] (0,0) -- (3,1) -- (-3,3) -- (0,0)}; 
        \end{tikzpicture}
        \]  
      \end{enumerate}
  \end{enumerate}


  % Stahl Ex 4.1.2 (p. 56)
\item Prove Proposition 4.1.2 in Stahl: The hyperbolic length of the
  Euclidean line segment joining the points $P=(a,y_1)$ and
  $Q=(a,y_2)$, where $0<y_1\leq y_2$, is $\ln(y_2/y_1)$. Do not use
  any later results that trivialize the problem.


  % Stahl Ex 4.4.5 (p. 62). 
\item Find the inversion $I_{C,k}$ that takes the bowed geodesic from
  $(-4,0)$ to $(2,0)$ to the bowed geodesic from $(10,0)$ to $(20,0)$.
  
  
  % Stahl Ex 4.4.10 (p. 62).
\item Given a hyperbolic geodesic $\gamma$ and a point $P$ on
  $\gamma$, describe a Euclidean method for constructing a hyperbolic
  geodesic through $P$ that is orthogonal to $\gamma$.


  % Stahl Ex 4.4.11 (p. 62). 
\item Given a hyperbolic geodesic $\gamma$ and a point $P$ not on
  $\gamma$, describe a Euclidean method for constructing a hyperbolic
  geodesic through $P$ that is orthogonal to $\gamma$.

 
\end{enumerate}

\end{document}
