% 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 Final Examination}}
\newcommand{\examdate}{June 11 2026}
\newcommand{\timelimit}{2 Hours}

\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
\renewcommand{\H}{\mathbb{H}}
\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[18] ({\bf Euclidean $\R^2$}) Consider the two points $P=(1,2),Q=(-1,-5)\in\R^2$ in the Euclidean plane. Solve the following parts:
\noaddpoints
\begin{parts}
\part[5] Find the Euclidean distance between $P$ and $Q$.
\vfill
\part[5] Let $M=\{(x,y)\in\R^2: x=y\}$  and consider the unique line $L\sse\R^2$ equidistant to $P$ and $Q$. Determine the image of $Q$ under the isometry $r_M\circ r_L$.
\vfill
\part[5] Find all the fixed points of the isometry $r_M\circ r_L$.
\vfill
\part[3] Show that there is no line $N\sse\R^2$ such that $r_N\circ r_M\circ r_L=\mbox{id}$.
\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[18] ({\bf $\Gamma$-Geometry for the cylinder}) Let $C=\R^2/\Gamma$ be the Euclidean cylinder, where $\Gamma=\langle t_{(1,0)}\rangle\sse\mbox{Iso}(\R^2)$ is the group generated by the translation
$$t_{(1,0)}:\R^2\lr\R^2.$$

\noaddpoints % If you remove this line, the grading table will show 20 points for this problem.
\begin{parts}
	\part[5] Draw the $\Gamma$-orbits $\Gamma P$ and $\Gamma Q$ in $\R^2$ of the two points $$P=(-1,4),Q=(3.25,-7.75)\in\R^2.$$
	\vfill
	\part[5] Compute the distance in $C$ from $\Gamma P$ to $\Gamma Q$.\\
	\vfill
	
	\part[5] Show that there are infinitely many lines in $C$ through $\Gamma P$ and $\Gamma Q$.\\
	\vfill
	
	\part[3] Consider the projections $\pi(L_1),\pi(L_2)$ to $C$ of the two lines
    $$L_1=\{x=0\}\sse \R^2,\quad L_2=\{47x+y=4\}\sse\R^2.$$ Explicitly find all the intersections points of $\pi(L_1)$ and $\pi(L_2)$ in $C$.\\
	\vfill
\end{parts}

\newpage
\addpoints
\question[18] ({\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] Consider the points $P=(1,0,0)\in S^2$ and $Q=\frac{1}{\sqrt{3}}(1,-1,1)$. Compute the spherical distance $d_{S^2}(P,Q)$ from $P$ to $Q$.

\vfill
\part[5] Let $R_{P,\pi/2},R_{Q,\pi/2}\in\mbox{Isom}(S^2)$ be the rotations of angle $\pi/2$ centered at $P$ and $Q$. Show that $R_{P,\pi/2}\circ R_{Q,\pi/2}$ is a rotation and find its center and angle.

\vfill
\part[5] Determine whether $R_{P,\pi/2}\circ R_{Q,\pi/2}$ is equal to $R_{Q,\pi/2}\circ R_{P,\pi/2}$.

\vfill
\part[3] Is there a spherical isometry $f\in\mbox{Isom}(S^2)$ such that $f\circ(R_{P,\pi/2},R_{Q,\pi/2})$ has no fixed points?

\vfill


\end{parts}

\newpage
\addpoints

\addpoints
\question[18] ({\bf Hyperbolic distances and lines in $\mathbb{H}^2$}) Let $P=i,Q=3+4i\in\H^2$ be points in the hyperbolic upper-half plane $\H^2$. Solve the following parts:
\noaddpoints
\begin{parts}
\part[5] Show that $L=\{z\in\H^2:|z-4|^2=17\}\sse\H^2$ is the unique hyperbolic line through the points $P$ and $Q$.
\vfill
\part[5] Compute the hyperbolic distance $d_{\H^2}(P,Q)$.
\vfill
\part[5] Find the unique hyperbolic line $M$ equidistant to $P$ and $Q$. 
\vfill
\part[3] Let $r_L,r_M\in\mbox{Isom}(\H^2)$ be the hyperbolic inversions along $L$ and $M$. Compute the image of $P$ and $Q$ under the composition $r_M\circ r_L$.
\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[18] ({\bf Hyperbolic isometries in $\mathbb{H}^2$}) Consider the map $f,g:\H^2\lr\H^2$
$$f(z)= \frac{\bar{z} + 9}{3\bar{z} - 5}.$$

\noaddpoints
\begin{parts}
\part[5] Show that $f$ is a hyperbolic isometry and it has no fixed points.
\vfill
\part[5] Let $g:\H^2\lr\H^2$ be a hyperbolic isometry such that
$$g\left(-\frac{21}{17} + \frac{16}{17}i\right)=i,\quad g\left(-\frac{33}{61} + \frac{64}{61}i\right)=2i,\quad g\left(-\frac{17}{13} + \frac{32}{13}i\right)=1+i.$$
Determine the image of the point $2+3i$ under the composition $f\circ g$.
\vfill
\part[5] Find a hyperbolic line $L\sse\H^2$ such that $f(L)=L$.\\
\vfill
\part[3] Show that there exists no $n\in\N$ such that $f^n=\mbox{id}$.
\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[10] 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] \small Two different lines in the twisted cylinder cannot intersect only at two points.\\
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill
	
	\part[2] Any orientation-reversing hyperbolic isometry must have a fixed point.\\
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill
	
	\part[2] A spherical isometry is determined by the image of three distinct points.\\
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill
	
	\part[2] For any two points $P,Q\in S^2$, there exists a unique line between them.\\
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill
	
	\part[2] Any hyperbolic triangle with vertices in $\H^2$ must have area less than $\pi$.\\
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill

    \part[2] There exists a regular hyperbolic hexagon whose angles are all $\pi/2$.
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill

    \part[2] For any two distinct hyperbolic lines $L,M\sse\H^2$, there exists a hyperbolic isometry $f:\H^2\lr\H^2$ with $f(L)=M$.
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill

    \part[2] If $L\sse\R^2$ is the line equidistant to $P,Q\in\R^2$, then any isometry fixing the points of $L$ must send $P$ to $Q$.
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill

    \part[2] If two hyperbolic lines $L,M$ do not intersect, then there exists a unique hyperbolic line that is perpendicular to both $L$ and $M$.
	
	(1) True. \qquad \qquad \qquad \qquad (2) False.
	\vfill
	
	
\end{parts}

\end{questions}
\end{document}
