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\pagestyle{fancy} \lhead{{\sf Homework 1 $|$ Due January 24 (Tuesday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 481 $|$ Topics in Geometry and Topology
    $|$ Spring 2012 $|$ M.~Macauley}}


\begin{document}

$\;$

\emph{Read}: Stahl, Chapters 1.1, 1.2, 1.3, 1.4, 2.1.

\begin{enumerate}


\item Let $X$ be a geometry, and let $f$ and $g$ be isometries of $X$.
\begin{enumerate}
\item Prove that $f$ is invertible and $f^{-1}$ is an isometry of $X$. 
\item Prove that $f\circ g$ is an isometry of $X$.
\end{enumerate}

\emph{For problems $2$--$6$, you are only allowed to use our analytic
  definition of Euclidean geometry and the results that we have
  derived from it.}


\item Prove that Euclidean length of a path is invariant under
  orientation-reversing change of parameter (recall that we did the
  orientation-preserving case in class).


\item Prove that the Euclidean angle between two paths at a given
  point of intersection is well-defined, that is, it is invariant
  under orientation-preserving change of parameter.


\item Prove that the translation $\tau_\v$ is a Euclidean isometry
  that preserves lines.


\item Prove that any two points in $\E^2$ can be moved to two points
  on the $x$-axis using a sequence of rotations around the origin and
  translations.
  

\item Prove that any pair of lines that meet at an angle $\theta$ can
  be moved by a Euclidean isometry onto any other pair of lines that
  meet at an angle $\theta$.

  \smallskip

\emph{For problems $7$--$8$, you may use both analytic and ruler-and-compass
methods.}


% Stahl Ex 2.2 (p. 50). Proof of Prop 2.2
\item Prove that if two rigid motions agree on two distinct points,
  then they agree everwhere on the line joining those two points.


\item Let $\tau_\v$ be a translation, and let $R_{C,\alpha}$ be a
  rotation around a point $C\in\E^2$. 
  \begin{enumerate}
  \item Describe, geometrically, the isometry $f:=\tau_\v\circ
    R_{C,\alpha}\circ\tau_\v^{-1}$. This is the unique map that makes
    the following diagram commute:
    \[
    \xymatrix{\E^2 \ar[rr]^{R_{C,\alpha}} \ar[d]_{\tau_\v} && \E^2 
      \ar[d]^{\tau_\v} \\ \E^2 \ar@{-->}[rr]^{f} && \E^2 \\ }
    \]
    What do $R_{C,\alpha}$ and $\tau_\v\circ
    R_{C,\alpha}\circ\tau_\v^{-1}$ have in common?
  \item Express $R_{C,\alpha}$ as a composition of translations and
    rotations around the origin.
  \end{enumerate}


\end{enumerate}

\end{document}
