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% Semester
\newcommand{\semester}{Fall, 2003}
% Name of the class.
\newcommand{\class}{Ma235--Calculus Laboratory}
% Title.
\newcommand{\labtitle}{Ants on a Doughnut}

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\noindent \semester  \hfill \class
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\begin{center}
{\bf \large \em \labtitle}\footnote{\url{http://www.math.unt.edu/~brand/projects/calc3/calc3.htm}}
\end{center}
\vspace{.1in}

Two ants are exploring a doughnut and notice that there are two regions on
the doughnut which have particularly good supplies of sugar and grease.
The two regions are as far apart on the doughnut as they can be.  The ants
must leave a path for all the other ants to follow.  In order to be
efficient the ants wish to find the shortest path between the good regions.

The ants notice that the doughnut is obtained by rotating the circle
$(x-2)^2+z^2=1$ (in the $xz$-plane) about the $z$-axis.  This puts the
doughnut in $xyz$-space.  The areas rich in sugar and grease are at the
positions $(3,0,0)$ and $(-3,0,0)$.

The first ant, Grant, suggests that the shortest route would be to follow
the path of $(3,0,0)$ as it rotates about the $z$-axis as in Figure
\ref{p1}  
%
\begin{figure}[h]
\center
%\includegraphics[width=2in]{p1.pdf}
\includegraphics[width=2in]{p1}
\caption{Staying on the outside of the dougnut.} \label{p1}
\end{figure}
%
Antonio, the
second ant suggests that the ``inside'' circle is smaller, so it would make
sense to follow the circle $(x-2)^2+z^2=1$ half way around so they are at
the ``small'' circle, follow it half way around to the point $(-1,0,0)$,
and then go around the circle from $(-1,0,0)$ to $(-3,0,0)$.  (See Figure
\ref{p2})
%
\begin{figure}[h]
\center
%\includegraphics[width=2in]{p2.pdf}
\includegraphics[width=2in]{p2}
\caption{Moving to the inside immediately, around the inside and
back to the outside.} \label{p2}
\end{figure}

After hearing Antonio's idea Grant thinks that it would be better not to go
directly around the circle $(x-2)^2+z^2=1$ before starting around the
$z$-axis rotation.  Grant says ``Why not go around both circles at the same
time?  Perhaps we could go around both circles simultaneously and save some
distance.''  (See Figure \ref{p3}.)
%
\begin{figure}[h]
\center
%\includegraphics[width=2in]{p3b.pdf}
\includegraphics[width=2in]{p3b}
\caption{Moving gradually around in both the $\alpha$ and $\beta$
parameters.} \label{p3}
\end{figure}

Antonio then suggests going around both at the same time part way until
they hit the `inside circle' on the $xy$-plane, around the inside circle
for a ways and finally around both until they arrive at $(-3,0,0)$.  Their
path could be symmetric about the $yz$-plane. (See Figure \ref{p4}.)
%
\begin{figure}[h]
\center
%\includegraphics[width=2in]{p4.pdf}
\includegraphics[width=2in]{p4}
\caption{In the first third of the path, moving gradually around in both
the $\alpha$ and $\beta$ parameters to a point on the inner circle, in the
second third of the path moving around the inner circle and for the third
third of the the path taking a path symmetric to that of the first third
back to the outside of the dougnut.} \label{p4} \end{figure}

Which route is shortest among those discussed?

Follow the steps below to help the ants find a short path between the two
regions.

\begin{enumerate}

 \item Think of each point on the torus (or doughnut) as being specified by
 two angles, say $\alpha$ and $\beta$.  One angle should measure how far
 the circle $(x-2)^2+z^2=1$ has revolved around the $z$-axes while the
 other should measure how far around the circle $(x-2)^2+z^2=1$ you rotate.
 Find formulas which give $x$, $y$, and $z$ in terms of the two angles.

 \item Make a graph of the paths originally suggested by the two ants in
 the $\alpha\beta$-plane.  Find the length of Grant's original path and
 Antonio's original path. (Remember, you must find the length in three
 space, not in the $\alpha\beta$-plane.) Which is shorter?  Would your
 answer change if the circle being rotated about the $z$-axis were moved
 closer to or further from the $z$-axis?

 \item Now make a graph in the $\alpha\beta$-plane of the second path
 suggested by Grant.  Find the length of this path.  You may have trouble
 computing the integral giving the exact path length.  If so, you may use
 Maple, or any other computer program (including one you write yourself) to
 estimate the integral which gives the path length you seek.

 \item Now make a graph in the $\alpha\beta$-plane of the second path
 suggested by Antonio.  Find the optimal point on the ``inside'' circle
 where the ants should aim before starting around the inside circle. (This
 will require some first semester calculus.  You are allowed to use the
 fact that $\partial_x\int_a^b f(x,t) dt = \int_a^b \partial_x f(x,t) dt$.)
 Assuming they aim for this point, what is the distance around the
 doughnut?  (You may wish to use the computer to approximate the optimal
 point and to compute the distance around.)

 \item Of the four paths suggested which is best?  Do you think there is a
 path shorter than any of the four you studied?

\end{enumerate}



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