\documentclass{amsart}
\usepackage{graphicx}

\title{Lab 2:  The Spirograph}

\begin{document}

\maketitle

In this lab, we use the Spirograph (a child's toy from Hasbro) to 
investigate parametric curves.  

Suppose you begin with a circle of radius $R$ centered
at the origin, $C$.  Take another circle of radius $r$, label
it $\hat{C}$, as shown in Figure
\ref{circles}.  A felt pen is connected to $\hat{C}$, and
it rotates about $C$.  The pen leaves a trace- a sample trace
is given in Figure \ref{examples}.

Here are the lab questions.  Your write up should be a discussion
that incorporates the answers- Don't just list them!  See the
sample write up for an example.

\begin{enumerate}
\item Come up with parametric equations, $x(t)$ and $y(t)$,
for the path of the pen in the Spirograph.  
\item Come up with some nice patterns!  You should try different
end values of $t$.  
\item Do all the curves end up being periodic
(in the sense that the curve is closed)?  Can you describe when
you will get closed curves?  (Experiment with different
values of $R$ and $r$).
\item Give the general formula for the arc length of the pen
in the spirograph.  Give a numerical value for the arc length
of a closed curve (from your previous example).
\item In the ``Conclusions'' section, you can bring in some
of the mathematical names of these curves.  You might do a 
little internet research to see if there is anything interesting
about them.
\end{enumerate}

\begin{figure}
\centering
\includegraphics[width=4.0in]{Circles}
\caption{The set up for the spirograph.}
\label{circles}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=4.0in,height=4.0in]{exampspiro1}
\includegraphics[width=4.0in,height=4.0in]{exampspiro2}
\label{examples}
\caption{Two examples.}
\end{figure}



\end{document}
