% 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 II}}
\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.
%
%
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%http://ksuweb.kennesaw.edu/~plaval/math4381/seqlimthm.pdf
\begin{questions}

% Question with parts
\addpoints
\question[20] ({\bf Isometries in $\R^2$}) Consider the three points $P=(0,0),Q=(1,0),R=(0,1)\in\R^2$ in the Euclidean plane. Let $f:\R^2\lr\R^2$ be an isometry such that $f(P)=(2,2)$, $f(Q)=(2,3)$ and $f(R)=(3,2)$. 
\noaddpoints
\begin{parts}
\part[5] Find the images $f(-1,0)$ and $f(8,2)$ of the points $(-1,0)$ and $(8,2)$ under the isometry $f$.
\vfill
\part[5] Prove that the isometry $f$ is not a translation, i.e. there exists no vector $(\alpha,\beta)\in\R^2$ such that $f=t_{(\alpha,\beta)}$.
\vfill
\part[5] Show that there exists no point $S\in\R^2$ such that $f(S)=S$.
\vfill
\part[5] Find a set of {\it at most} three reflection $\{\overline{r}_{L_1},\overline{r}_{L_2},\overline{r}_{L_3}\}\in\mbox{Iso}(\R^2)$ such that $f$ is a composition of these reflections.
\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.

\addpoints
\newpage
\question[20] ({\bf $\Gamma$-Geometry for the Klein Bottle}) Let $K=\R^2/\Gamma$ be the Euclidean Klein Bottle, where $\Gamma=\langle t_{(0,1)},\overline{r}\circ t_{(1,0)}\rangle\sse\mbox{Iso}(\R^2)$.

\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 following points: $$P=(0,0),Q=(0.5,2),R=(1,-5),S=(3,-232)\in\R^2.$$
	\vfill
	\part[5] Find a fundamental domain $D_\Gamma\sse\R^2$ which is {\it not} a square.\\
	\vfill
	
	\part[5] Consider the lines
	$$L=\{(x,y)\in K:x=2y\},\quad M=\{(x,y)\in K:x=0\}.$$
	Find {\it all} the intersection points $L\cap M$.\\
	\vfill
	
	\part[5] Consider the line $N=\{(x,y)\in K:x=\pi\cdot y\}$. Is the number of intersection points $M\cap N$ finite or infinite ?\\
	\vfill
\end{parts}
\newpage
\addpoints
\question[20] ({\bf The Cylinder}) In this problem, {\it all} points and lines are considered in the cylinder $C=\R^2/\Gamma$, where $\Gamma=\langle t_{(1,0)}\rangle\sse\mbox{Iso}(\R^2)$. Solve the following parts:\\
\noaddpoints

\begin{parts}
	\part[5] Consider the points $P=(0.5,0),Q=(0.3,0.2),R=(5.9,-0.2)\in M$. Find an isometry $f:C\lr C$ such that
	$$f(P)=(0.7,0),\quad f(Q)=(0.5,-0.2),\quad f(R)=(6.1,0.2).$$
\vfill	

	\part[5] Find infinitely many distinct lines $\{L_i\}\sse C$, $i\in\N$, which contain $P,Q$, i.e. $P,Q\in L_i$, for all $i\in\N$.
	
	\vfill
	\part[5] Let $t_{(0,\pi)}:C\lr C$ be a vertical translation, and $H=\langle t_{(0,\pi)}\rangle$ the group of isometries of $C$ it generates. Does the $H$-orbit of the point $R\in C$ have limit points in the cylinder $C$ ? (Justify your answer.)
	
	\vfill
	
	\part[5] Consider the group $A=\langle t_{(0,\sqrt{2})},t_{(0,1)}\rangle$ as a subgroup of the group of isometries of $C$. Prove that the $A$-orbit of $P$ inside the cylinder $C$ has limit points.
	
	\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] Compute the distance between $(1,0,0)$ and $(0,0,1)$.

\vfill
\part[5] Determine the set of points in $S^2$ whose distance to $(0,1,0)$ equals their distance to $(0,0,1)$.


\vfill
\part[10] Let $E=\{z=0\}\sse S^2$ be the equator. Find an orientation-reversing isometry $f:S^2\lr S^2$ such that $f(E)=E$ but $f$ has no fixed points on the equator.

\vfill


\end{parts}

\newpage
\addpoints

\addpoints
\question[20] For each of the five sentences below, circle the {\bf unique} correct answer. 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] Let $(0,0),(0.5,0)\in C$ be two points in the cylinder. The set of points equidistant to $(0,0)$ and $(0.5,0)$ consists of exactly:\\
	
	(1) A line,\qquad \qquad(2) Empty\qquad \qquad(3) Two lines\qquad \qquad(4) Infinite Lines
	\vfill
	
	\part[2] Two lines $L,M\sse T^2$ in the two torus must have:\\
	
	(1) Finitely Many Intersection Points  \qquad (2) Infinitely Many Intersection Points\\ \\
	(3) No Intersection Points. \qquad\qquad (4) None of the other answers.
	\vfill
	
	\part[2] A non-trivial subgroup $\Gamma\sse\mbox{Iso}(\R^2)$ must:\\
	
	(1) Contain a non-trivial translation, \qquad (2) Be generated by at most two elements,\\ \\
	(3) Be fixed point free, \qquad (4) Contain a product of reflections.
	\vfill
	
	\part[2] There exists a unique isometry which fixes\\
	
	(1) Three collinear points \qquad (2) Three non-collinear points\\
	
	(3) Four collinear points \qquad (4) The origin.
	\vfill
	
   	\part[2] Let $f:S^2\lr S^2$ be an isometry. Then\\
		
(1) $f$ cannot have fixed points,\\

(2) $f$ is a product of at most three reflections,\\

(3) $f$ is a product of at most two reflections,\\

(4) the fixed point set of $f$ must be two antipodal points.
	\vfill
\end{parts}

\end{questions}
\end{document}
