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\pagestyle{fancy} \lhead{{\sf Homework 2 $|$ Due January 31 (Tuesday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 481 $|$ Topics in Geometry and Topology
    $|$ Spring 2012 $|$ M.~Macauley}}


\begin{document}

$\;$

\emph{Read}: Stahl, Chapters 2.2, 2.3, 2.4, 2.5, 3.1.

\begin{enumerate}
  
\item In this problem, use only the analytic definition of Euclidean
  geometry. Let $\v$ be a unit vector in $\R^2$, and define
  $r_\v(\x)=\x-2(\x\cdot\v)\v$.
  \begin{enumerate}
  \item Prove (analytically) that $r_\v$ is a Euclidean isometry.
  \item Prove that $r_\v$ is the reflection in a line $m$ through the
    origin, and find an equation for that line (both in $y=mx+b$, and
    in parametric form).
  \item For an arbitrary $\alpha\in\R$, describe the isometry
    $f:=\tau_{\alpha\v}\circ r_\v\circ\tau_{\alpha\v}^{-1}$
    geometrically, and prove your description. This is the unique map
    that makes the following diagram commute:
    \[
    \xymatrix{\E^2 \ar[rr]^{r_\v} \ar[d]_{\tau_{\alpha\v}} && \E^2
      \ar[d]^{\tau_{\alpha\v}} \\ \E^2 \ar@{-->}[rr]^{f} && \E^2 \\ }
    \]
    What do $r_\v$ and $\tau_{\alpha\v}\circ
    r_\v\circ\tau_{\alpha\v}^{-1}$ have in common?
  \end{enumerate}


\emph{In the following three problems, you will prove a series of
  results stated in Chapter 2 of Stahl. For each problem, you are free
  to use results in Stahl that appear before it, provided you clearly
  state and cite them. The notation $\rho_m$ denotes the reflection of
  $\E^2$ across the line $m$, and $\tau_{AB}$ denotes the translation
  of $\E^2$ that carries $A$ to $B$.}


  % 2.2. Stahl Ex 2.2.8 (p. 34).
\item Prove Proposition 2.2.7 in Stahl: Let $m$ and $n$ be two
  parallel straight lines. Let $AB$ be a line segment that first
  intersects $m$ and then $n$, that is perpendicular to both, whose
  length is twice the distance between $m$ and $n$. Prove that
  $\rho_n\circ\rho_m=\tau_{AB}$.


  % 2.3. Stahl Ex 2.3.6 (p. 37).
\item Prove Lemma 2.3.1 in Stahl: Let the line segment $AB$ be
  perpendicular to the line $m$. Prove that
  $\rho_m\circ\tau_{AB}=\rho_n$, where $n$ is a line parallel to $m$.
  
  
  % 2.4. Stahl Ex 2.3.7 & 2.3.8 (p. 37). Finish proofs of Prop 2.3.2 & 2.3.3
\item Prove the second parts of Propositions 2.3.2 and 2.3.3 in Stahl:
  Let $\rho$ be any reflection, $\tau$ be any translation, and $R$ be
  any rotation of $\E^2$. Prove that $\rho\circ\tau$ and $\rho\circ R$
  are glide reflections. (Note: The facts that $\tau\circ\rho$ and
  $R\circ\rho$ also glide reflections constitute the first part of
  Propositions 2.3.2 and 2.3.3.)
  

  % 2.5.
\item In this problem, you may use the classification of Euclidean
  isometries as described in Theorem 2.4.3 of Stahl. 
  \begin{enumerate}
  \item Let $m$ be a line in $\E^2$. Describe all of the
    isometries of $\E^2$ that fix every point of $m$, and prove that
    your description is complete.
  \item Let $P$ be a point of $\E^2$. Describe all of the
    isometries of $\E^2$ that fix $P$, and prove that your description
    is complete.
  \end{enumerate}


% Prop 2.1.3 of the supplemental notes
\item Recall that the circle inversion in polar coordinates is the map
  $I_{O,k}\colon\E^2\setminus O$ given by
  $I_{O,k}(r,\theta)=(k^2/r,\theta)$. Now, let
  $f(r,\theta)\colon(0,\infty)\times[0,2\pi)$ be an arbitrary
  function, and define the sets $\Gamma_0:=\{(r,\theta)\mid
  f(r,\theta)=0\}$ and $\Gamma_1:=\{(r,\theta)\mid
  f(k^2/r,\theta)=0\}$. Prove that $I_{O,k}(\Gamma_0)=\Gamma_1$.


\newpage

$\;$

\item Theorem 3.1.6 of Stahl says that $I_{C,k}$ is a conformal
  mapping of $\E^2\setminus C$, that is, it preserves angles between
  paths. In this problem, we will prove this statement in a more
  rigorous and elegant fashion, using only the analytic methods we
  developed in class.
  \begin{enumerate}
  \item Prove that for any angle $\theta$, the matrix
    \[
    B_\theta=\left(\!\begin{array}{cc}\cos\theta & \sin\theta \\ \sin\theta
    & -\cos\theta \end{array}\!\right)
    \]
    is orthogonal. 
  \item Prove that, at any given point $(x,y)\neq(0,0)$, the inversion
    $I'_{O,k}(x,y)$ is a scalar multiple of the matrix $B_\theta$ for
    some value of $\theta$.
  \item Prove that $I_{O,k}$ preserves angles between paths in
    $\E^2\setminus O$.
  \item For a vector $\v\in\R^2$, describe the transformation
    $\tau_\v\circ I_{O,k}\circ\tau_\v^{-1}$ geometrically, and prove
    your descrpition. Draw a commutative diagram that shows how these
    maps are related. (Be careful: The domains of these maps are not
    quite all of $\E^2$!)
  \item Prove that the transformation $I_{C,k}$ preserves angles
    between paths in $\E^2\setminus C$.
  \end{enumerate}



\end{enumerate}

\end{document}
