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\lhead{Math 4441 Activity 4}
\rhead{<your name(s)>}


\begin{document}
\title{Math 4441 Activity 4}
\author{<your name(s)>}
\date{September 21, 2022}
\maketitle
\vspace{5em}

\setcounter{section}{4}

\begin{exercise}
Draw a pair of closed curves, one of which is convex, and one of which is not.  Then draw a pair of curves which are not closed, one convex and one not.  For the not-convex curves, draw a tangent line that demonstrates the failure of convexity.
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\begin{exercise}
Draw a simple, regular, planar curve whose planar curvature has constant sign, but which is not convex.  Be sure to draw a tangent line which demonstrates the failure of convexity!
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\begin{exercise}
Draw a closed, regular, planar curve whose planar curvature has constant sign, but which is not convex.  Be sure to draw a tangent line which demonstrates the failure of convexity!
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\begin{exercise}\label{activity-4:parallel-lines}
What can you say about the tangent lines $\ell_{P_2}$ and $\ell_{P_3}$ in relation to $\ell_{P_1}$?  (Be sure to prove your claim!)
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\begin{exercise}
Based on your observation in Exercise~\ref{activity-4:parallel-lines}, explain why we can conclude that there's some way of choosing $i$ and $j$ to make the equation $\vec{t}(s_i)=\vec{t}(s_j)$ true.
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\begin{exercise}\label{activity-4:angular-rotation}
Based on what we've done so far, prove that either $\theta(s_2)=\theta(s_1)$ or $\theta(s_2)=\theta(s_1)\pm 2\pi$.\\
\emph{Hints: Use the rotation index theorem and the fact that $\theta'=k$ is non-negative.}
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\begin{exercise}
Use the observation that $\theta$ is constant on an arc connecting $P_1$ to $P_2$ to obtain a contradiction, completing the proof.
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\begin{exercise}
Explain why the tangent lines to $\vec{\alpha}(s_1)$, $\vec{\alpha}(s_2)$, and $\vec{\alpha}(s_3)$ cannot be pairwise distinct.\\
\emph{Hint: If they were, one of them would have to be in the middle.}
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\begin{exercise}
Show that every point of the line segment $\overline{AB}$ lies on $\vec{\alpha}$.
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\end{document}