\magnification=\magstep1
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\font\bigrm=cmr17
\font\medrm=cmr10 scaled\magstep2

\def\csch{{\rm csch}}
\def\sech{{\rm sech}}

\centerline{\bigrm Spirographs}
\bigskip

\noindent
{\bf Purpose:}
The purpose of this project is to reinforce your understanding of
parametric equations.
\bigskip

\noindent
{\bf Procedure:}  Read the outline below and derive the required formulas.
After completing the derivation test your answer by using Mathematica
(or some other graphing program or calculator) to
plot the graphs.  Then write your
results {\bf neatly}.  Be sure to include relevant
graphs
in what you turn in. Your grade will be determined
in part by your presentation.  Include a careful explanation of how you
derived the formulas with relevant pictures.
\bigskip

\noindent
{\bf Project Outline:}
Katie's little brother Kelly has a spirograph which he uses to make
pretty designs.  There are two problems with the spirograph.  After
Kelly gets everything all lined up and starts to draw the design,
either the gear slips or else the pen quits in the middle of the
design.  One day while Katie was working on the computer Kelly had
almost completed a beautiful picture when the gear slipped.  Kelly
started to cry.
Katie promised Kelly that if he would tell Katie which
gear he was using then Katie could write a computer program to print
out the design.  Katie promised to have a program written within two
weeks that would print out any of the designs Kelly would want which
involve two circles.  That night after Kelly went to bed Katie realized
that she needed some sort of an equation so she could graph it.

The next day Kelly asked Katie if she knew how long the curve traced by
the pen is when the moving circle goes around the fixed circle once.
Not wanting to look stupid to her little brother, Kelly said that she
could compute the length if Kelly told her the radii of both circles and
where he placed the pen,
but she didn't have time right now to do it.  After
she finishes the program she will work it out.  Kelly said he wanted to
know the distance when both radii were 1, the outer circle moves and the
pen is placed on the edge of the circle.  He then asked if
placing the pen in the center of the circle always
gives the shortest path when
the outer moving circle goes around exactly once,
regardless of the radii of
the circles.

After thinking about the problem Katie started to panic. She
then remembered that you were taking calculus so she called and asked
for help.  It is your job to find a parametric equation whose graph
gives the desired design. Then, based on the equations, compute the
desired path length and either verify
Kelly's guess that the center always gives the minimum path length or
else show that it doesn't always.

In case you are not familiar with a spirograph, the designs are created
by rolling one circle either inside or outside
a fixed circle with the pen attached to the rolling circle.  See Figure~1.
Note that the pen's positon can be fixed anywhere inside the rotating
circle.

You should show the derivation of the equation for both cases.
To check your answer, plot your equations using Mathematica, a graphing
calculator, or another computer software package.
When determining arc length it will be possible to calculate
the integral exactly, at least in the case where Kelly wants the answer.
You may wish to do some experimentation using numerical integration to
see if Kelly's guess is correct.
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}



\bye