% 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.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 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}}
\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}

% Question with parts
\addpoints
\question[20] ({\bf Rotations in $\R^2$}) Consider the two points $P=(0,0),Q=(1,0)\in\R^2$ in the Euclidean plane. Solve the following parts:
\noaddpoints
\begin{parts}
\part[5] Let $R_{P,\pi/2}$ be a rotation of angle $\pi/2$ centered at $P$. Compute the image $R_{P,\pi/2}(3,3)$ of the point $(3,3)\in\R^2$ under the isometry $R_{P,\pi/2}$.
\vfill
\part[5] Let $R_{Q,-\pi/2}$ be a rotation of angle $-\pi/2$ centered at $Q$. Compute the image $R_{Q,-\pi/2}(4,5)$ of the point $(4,5)\in\R^2$ under the isometry $R_{Q,-\pi/2}$.
\vfill
\part[5] Let $(x,y)\in\R^2$ be any point. Where does $(x,y)\in\R^2$ get send under the composition $R_{Q,-\pi/2}\circ R_{P,\pi/2}$ ?
\vfill
\part[5] Show that $R_{Q,-\pi/2}\circ R_{P,\pi/2}=t_{(1,1)}$.
\vfill
\end{parts}

% 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 $\Gamma$-Geometry for the 2-Torus}) Let $T^2=\R^2/\Gamma$ be the Euclidean Torus, where $\Gamma=\langle t_{(0,1)},t_{(1,0)}\rangle\sse\mbox{Iso}(\R^2)$ is the group generated by the two translations
$$t_{(0,1)},t_{(1,0)}:\R^2\lr\R^2.$$

\noaddpoints % If you remove this line, the grading table will show 20 points for this problem.
\begin{parts}
	\part[5] Draw the $\Gamma$-orbits of the two points $P=(2,3),Q=(0.5,-7.5)\in\R^2$.
	\vfill
	\part[5] Find a fundamental domain $D_\Gamma\sse\R^2$ which contains $P\in\R^2$.\\
	\vfill
	
	\part[5] Consider $P=(2,3),Q=(0.5,-7.5)\in\R^2/\Gamma$ as points in the 2-torus. Show that the line $\{(x,y)\in T^2:x=y\}\sse T^2$ contains both $P$ and $Q$.\\
	\vfill
	
	\part[5] Find {\it all} lines $L\sse T^2$ such that $P,Q\in L$.\\
	\vfill
\end{parts}
\newpage
\addpoints
\question[20] ({\bf Geometry in the Twisted Cylinder}) In this problem, {\it all} points and lines are considered in the twisted cylinder $M=\R^2/\Gamma$, where $\Gamma=\langle t_{(1,0)}\circ \overline{r}\rangle\sse\mbox{Iso}(\R^2)$. Solve the following parts:\\
\noaddpoints

\begin{parts}
	\part[5] Consider the points $P=(0,0),Q=(0.9,0.2),R=(5.9,-0.2)\in M$. Find the three distances $d(P,Q),d(P,R),d(Q,R)\in M$.
\vfill	

	\part[5] Find the intersection points between the line $\{(x,y)\in M:x=0.5\}\sse M$ and the line $\{(x,y)\in M:x=-y\}\sse M$.
	
	\vfill
	\part[5] Find two lines $K,L\sse M$ such that $|L\cap K|=2$.
	
	\vfill
	
	\part[5] Show that given two points $S,T\in M$ in the complement of the line $H=\{(x,y)\in M:y=0\}\sse M$, there exists a continuous path $\gamma\sse M$ from $S$ to $T$ such that $|H\cap\gamma|=0$.
	
	\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.

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

\begin{parts}
\part[5] Show that there exists a unique line $L\sse S^2$ containing the points $(1,0,0)$ and $(0,1,0)$.

\vfill
\part[5] Prove that there are infinitely many lines in $S^2$ containing the points $(1,0,0)$ and $(-1,0,0)$.

\vfill
\part[10] Let $R_1:S^2\lr S^2$ be the rotation with center $(1,0,0)$ and angle $\pi/2$, and $R_2:S^2\lr S^2$ be the rotation with center $(0,0,1)$ and angle $\pi/2$. Determine whether the isometry $R_1\circ R_2$ is equal to $R_2\circ R_1$.

\vfill


\end{parts}

\newpage
\addpoints

\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] Two lines $K,L\sse T^2$ cannot intersect at more than one point.\\
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill
	
	\part[2] Let $\Gamma\sse\mbox{Iso}(\R^2)$ be generated by translations. Then an isometry $g\in\Gamma$ cannot have fixed points.\\
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill
	
	\part[2] The composition of an even number of reflections cannot be a reflection.\\
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill
	
	\part[2] A glide reflection admits infinitely many fixed points.\\
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill
	
	\part[2] For any glide reflection $\overline{r}_1\in\mbox{Iso}(\R^2)$, there exists a glide reflection $\overline{r}_2\in\mbox{Iso}(\R^2)$ such that $\overline{r}_2\circ\overline{r}_1=\mbox{Id}$.\\
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill

    \part[2] An isometry $f:S^2\lr S^2$ is uniquely determined by the image of three points.
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill

    \part[2] There are no isometries in the M\"obius band except for the identity.
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill

    \part[2] The product of two rotations in $S^2$ is necessarily a rotation.
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill

    \part[2] Any isometry $f:S^2\lr S^2$ can be written as the product of at most three rotations.
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill
	
	
\end{parts}

\end{questions}
\end{document}
