% Exam Template for UMTYMP and Math Department courses
%
% Using Philip Hirschhorn's exam.cls: http://www-math.mit.edu/~psh/#ExamCls
%
% run pdflatex on a finished exam at least three times to do the grading table on front page.
%
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% These lines can probably stay unchanged, although you can remove the last
% two packages if you're not making pictures with tikz.
\documentclass[12pt]{exam}
\RequirePackage{amssymb, amsfonts, amsmath, latexsym, verbatim, xspace, setspace}
\RequirePackage{tikz, pgflibraryplotmarks}

% By default LaTeX uses large margins.  This doesn't work well on exams; problems
% end up in the "middle" of the page, reducing the amount of space for students
% to work on them.
\usepackage[margin=1in]{geometry}


% Here's where you edit the Class, Exam, Date, etc.
\newcommand{\class}{University of California Davis}
\newcommand{\term}{Euclidean Geometry MAT 141}
\newcommand{\examnum}{\color{red}{Sample Midterm Examination III}}
\newcommand{\examdate}{May 1 2026}
\newcommand{\timelimit}{50 Minutes}

\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\sse}{\subseteq}
\newcommand{\lr}{\longrightarrow}

% For an exam, single spacing is most appropriate
\singlespacing
% \onehalfspacing
% \doublespacing

% For an exam, we generally want to turn off paragraph indentation
\parindent 0ex

\begin{document} 

% These commands set up the running header on the top of the exam pages
\pagestyle{head}
\firstpageheader{}{}{}
\runningheader{}{\examnum\ - Page \thepage\ of \numpages}{\examdate}
\runningheadrule

\begin{flushright}
\begin{tabular}{p{2.8in} r l}
\textbf{\class} & \textbf{Name (Print):} & \makebox[2in]{\hrulefill}\\
\textbf{\term} & \textbf{Student ID (Print):} &
\makebox[2in]{\hrulefill}\\ \\
\textbf{\examnum} && \textbf{\examdate}\\
\textbf{Time Limit: \timelimit} &
\end{tabular}\\
\vspace{1cm}
\end{flushright}
\rule[1ex]{\textwidth}{.1pt}

\vspace{1cm}
This examination document contains \numpages\ pages, including this cover page, and \numquestions\ problems.  You must verify whether there any pages missing, in which case you should let the instructor know. {\bf Fill in} all the requested information on the top of this page, and put your initials on the top of every page, in case the pages become separated.\\

You may \textit{not} use your books, notes, or any calculator on this exam.\\

You are required to show your work on each problem on this exam.  The following rules apply:\\

\begin{minipage}[t]{3.7in}
\vspace{0pt}
\begin{itemize}

\item[(A)] \textbf{If you use a lemma, proposition or theorem which we have seen in the class or in the book, you must indicate this} and explain why the theorem may be applied.

\item[(B)] \textbf{Organize your work}, in a reasonably neat and coherent way, in the space provided. Work scattered all over the page without a clear ordering will receive little credit.  

\item[(C)] \textbf{Mysterious or unsupported answers will not receive full
credit}.  A correct answer, unsupported by calculations, explanation,
or algebraic work will receive little credit; an incorrect answer supported
by substantially correct calculations and explanations will receive
partial credit.


\item[(D)] If you need more space, use the back of the pages; clearly indicate when you have done this.
\end{itemize}

Do not write in the table to the right.
\end{minipage}
\hfill
\begin{minipage}[t]{2.3in}
\vspace{0pt}
%\cellwidth{3em}
\gradetablestretch{2}
\vqword{Problem}
\addpoints % required here by exam.cls, even though questions haven't started yet.	
\gradetable[v]%[pages]  % Use [pages] to have grading table by page instead of question

\end{minipage}
\newpage % End of cover page

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% See http://www-math.mit.edu/~psh/#ExamCls for full documentation, but the questions
% below give an idea of how to write questions [with parts] and have the points
% tracked automatically on the cover page.
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%http://ksuweb.kennesaw.edu/~plaval/math4381/seqlimthm.pdf
\begin{questions}

% If you want the total number of points for a question displayed at the top,
% as well as the number of points for each part, then you must turn off the point-counter
% or they will be double counted.
\newpage
\addpoints
\question[20] ({\bf Properties of $\Gamma\sse\mbox{Iso}(\R^2)$}) Solve the following parts:


\noaddpoints % If you remove this line, the grading table will show 20 points for this problem.

\begin{parts}



\part[5] Let $L=\{(x,y)\in\R^2:x=y\}$ and $M=\{(x,y)\in\R^2:x=6\}$. Find an element $g\in\Gamma=\langle \overline{r}_{L},\overline{r}_{M}\rangle$ which has a {\it unique} fixed point.

\vfill
\part[5] Draw the $\Gamma$-orbit of the point $(0,0)$ where $\Gamma:=\langle t_{(2,3)},t_{(-1,0)}\rangle$.
\vfill

\part[5] Give an instance of a line $L\in\R^2$ such that its image under the quotient $\R^2\lr\R^2/\Gamma$ is finite, where $\Gamma:=\langle t_{(2,3)},t_{(-1,0)}\rangle$ as in $(b)$.
\vfill

\part[5] Show that any non-trivial isometry in $\Gamma:=\langle t_{(2,3)},\overline{r}\circ t_{(1,0)}\rangle$ cannot have fixed points.


%\vfill
%\part[5] Find two elements $g_1,g_2$ in the group
%$$\Gamma:=\langle t_{(-4,6)},t_{(-3,9)},t_{(5,-15)},t_{(2,-3)},t_{(-1,3)},t_{(1,-1.5)}\rangle$$ which generate $\Gamma$, i.e. such that $\Gamma=\langle g_1,g_2\rangle$.



\vfill

\end{parts}

\newpage
\addpoints
\question[20] ({\bf Reflections in $\R^2$}) Consider the two lines $L_0=\{y=0\},L_1=\{x=y\}\sse\R^2$ and the two lines $M_0=\{x=y+1\},M_1=\{x=-y+1\}\sse\R^2$.


\noaddpoints % If you remove this line, the grading table will show 20 points for this problem.

\begin{parts}
\part[5] Show that the only fixed point of the isometry $\overline{r}_{L_1}\circ\overline{r}_{L_0}$ is $(0,0)$.

\vfill
\part[5] Prove that the isometry $\overline{r}_{M_1}\circ\overline{r}_{M_0}\circ\overline{r}_{L_1}\circ\overline{r}_{L_0}$ is a rotation.

\vfill
\part[5] Show that there exist two lines $N_0,N_1\sse\R^2$ such that $$\overline{r}_{M_1}\circ\overline{r}_{M_0}\circ\overline{r}_{L_1}\circ\overline{r}_{L_0}=\overline{r}_{N_1}\circ\overline{r}_{N_0}.$$

\vfill
\part[5] Find the image of a point $(x,y)\in\R^2$ under the isometry $\overline{r}_{M_0}\circ\overline{r}_{L_1}$.

\vfill


\end{parts}


\newpage

\addpoints
\question[20] ({\bf Spherical geometry}) Consider the 2-sphere
$$S^2:=\{(x,y,z)\in\R^3:~x^2+y^2+z^2=1\},$$
endowed with the spherical distance. Solve the following parts:
\noaddpoints % If you remove this line, the grading table will show 20 points for this problem.
\begin{parts}

	\part[5] Consider the point $P=\frac{1}{\sqrt{3}}(1,1,1)$. Compute its spherical distance to each of the points $Q_1=(1,0,0),Q_2=(0,1,0)$ and $Q_3=(0,0,1)$.
	
	
	\vfill

    \part[5] Let $L\sse S^2$ be the unique line containing $P$ and $Q_1$, show that the point $\frac{1}{\sqrt{2}}(0,1,1)$ belongs to $L$.\\
	
	\vfill
	
	\part[5] Let $E$ the unique line containing $Q_1$ and $Q_2$. Find the image of the unique line through $Q_1$ and $Q_3$ under the composition $\overline{r}_{L}\circ \overline{r}_{E}$.\\
	
	\vfill
	
	\part[5] Find the fixed points of the isometry $f$ obtained by first applying $\overline{r}_{L}\circ \overline{r}_{E}$ and then applying the reflection $(x,y,z)\lr (-x,y,z)$.\\
	
	\vfill

	
\end{parts}

\newpage
\addpoints
\question[20] For each of the ten sentences below, circle whether they are {\bf true} or {\bf false}. You do {\it not} need to justify your answer.\\
\noaddpoints % If you remove this line, the grading table will show 20 points for this problem.
\begin{parts}

	\part[2] For any pair of points $P,Q\in K$ in the Klein bottle, there are infinitely many distinct lines $L\sse K$ containing $P,Q\in K$.
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill
	
	\part[2] There exist rotations $R_{P,\theta},R_{Q,\phi}\in\mbox{Iso}(\R^2)$ such that the composition $R_{P,\theta}\circ R_{Q,\phi}$ is {\it not} a rotation.\\
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill
	
	\part[2] Two lines $L,K\sse M$ in the twisted cylinder either intersect $0$,$1$ or infinitely many times.\\
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill
	
	\part[2] Let $\Gamma\sse\mbox{Iso}(\R^2)$ be generated by a finite number of translations. Then there exists a fundamental domain $D_\Gamma\sse\R^2$ of finite area.\\
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill

	\part[2] The quotient of the isometry $t_{(0,1)}:\R^2\lr\R^2$ gives a well-defined isometry in the twisted cylinder $\R^2/\langle t_{(1,0)}\circ\bar{r}\rangle$.\\
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill

    \part[2] There are no parallel lines in the cylinder.
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill

    \part[2] Every orientation-preserving isometry of the 2-sphere is a rotation.
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill

    \part[2] Every orientation-reserving isometry of the 2-sphere has a fixed point.
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill

    \part[2] There are parallel lines in the 2-sphere.
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill

    \part[2] The spherical distance between two points in the 2-sphere is the same as the Euclidean distance between these two points, as computed in $\R^3$.
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill
	
\end{parts}
\end{questions}
\end{document}
