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\magnification=\magstep1
\nopagenumbers
\centerline{\bigrm Ants on a Doughnut}
\bigskip

\noindent
{\bf Purpose.}  In this project you will use calculus to solve a
  geometry problem.  This project serves as an introduction to
  differential geometry.
\medskip

\noindent
{\bf The Problem.}  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.
  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 1.)

  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 2.)

 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~3.)

  Which route is shortest among those discussed?
\medskip

\noindent
{\bf Procedure.}  Follow the steps below to help the ants find a short
  path between the two regions.
\smallskip
 \item{1.}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{2.}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{3.}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 Mathematica, Maple, or any
          other computer program (including one you write yourself)
          to estimate the integral which gives
          the path length you seek.
 \item{4.}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 $D_x\int_a^b f(x,t) dt = \int_a^b D_x f(x,t) dt$.)
          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{5.}Of the four paths suggested which is best?
          Do you think there is a path shorter than any of the four you
          studied?

\vbox{
\vskip 7.25true in
\special{psfile="tori.eps"}
}

\bye