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\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.  

We'll simplify the problem to a small circle rotating about
a larger, fixed, circle.  There is a pen attached to the smaller circle
(denoted by an asterisk in Figure \ref{circles}).  Define
$R$ to be the radius of the large circle, $r$ is the radius
of the small circle, and $r+h$ is the ``radius'' to the pen.
Furthermore, we have two angles- Let $t$ be the angle for the
large circle and $\theta$ is the angle for the smaller circle.

Here are the lab questions.  Your write up should be a discussion
that incorporates the answers- Don't just list them!  See the
{\bf sample write} (on the class website) up for an example.  You
might also see the grading sheet to get an idea of how the 
lab will be graded (also online).

\begin{enumerate}
\item Come up with parametric equations, $x(t)$ and $y(t)$,
for the path of the pen in the Spirograph.  There are multiple
ways of defining $t$- For this lab, let us assume that $t$ 
is the angle coming from the large, fixed, circle.  Here 
are some notes that might help you get started:
\begin{enumerate}
\item Given a circle of radius $r$, the $(x,y)$ coordinates
of a point on its boundary are $(r\cos(\theta), r\sin(\theta))$,
where $\theta$ is the central angle (measured from the positive
$x-$axis).
\item The arc length $s$ from a circle of radius $r$, and
measured from a central angle $\theta$:  $s=r\theta$
\item Try first writing the coordinates of the rotating
circle.  Define $t$ as the angle for the fixed circle,
and $\theta$ be the angle for the small circle.  What
are the coordinates of the pen?
\end{enumerate}
\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 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}



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