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\centerline{Mathematics 392--Seminar in Computational Commutative Algebra}
\centerline{Lab Day 3/Midterm Problem Set}
\centerline{October 13, 2006}
\bigskip
\noindent
{\it General Information and Ground Rules}
\bigskip
Today's lab session, and the problems for the Midterm Problem Set
will focus on two applications of elimination: the {\it implicitization}
problem for parametric surfaces, and finding the {\it envelope}
of a family of curves in the plane.  These topics are discussed in 
sections 3 and 4 of Chapter 3 in ``IVA''.  You should submit
solutions for the First Problems/Examples I, II as well 
as problems III, IV (Maple is not needed for this one), and 
V for the Midterm Problem Set, which is due on Friday, October 20.  
\bigskip
This is an open book, open notes, open Swords 335 assignment.
You must work {\it independently} on these problems, and you
may not consult with other faculty members (e.g. Gareth Roberts)
or students.  You may consult other sources in the library, 
on the Internet, etc. but if you find something relevant 
there, please {\it footnote it}.  When you have finished your 
solutions, please write out a short statement that you have 
abided by these ground rules, sign it, and return it with your 
solutions.
\bigskip
\noindent
{\it First Problems/Examples}
\bigskip
\noindent
I.  Consider the parametric surface in ${\bf C}^3$ defined by 
$$\eqalign{x &= uv + v \cr
           y &= u^2 - 2v\cr
           z &= v^2 + u - v\cr}$$
\item{A.}  Compute a {\it lex} Gr\"obner basis for the 
ideal 
$$I = \langle x - uv - v, y - u^2 + 2v, z - v^2 - u + v\rangle$$
with the variables arranged to eliminate $u,v$.  
\item{B.}  Use your answer to part A to find a variety 
$V \subset {\bf C}^3$ containing
the image of the parametrization mapping.
\item{C.}  Use the Extension Theorem from class on Wednesday
to prove that every point of $V$ is in the image of the 
parametrization mapping over ${\bf C}$.  (Hints:  This is equivalent to showing
that every partial solution $(x,y,z)$ in $V(I_2) \subset {\bf C}^3$ 
extends to a full solution $(x,y,z,u,v)$ in $V(I) \subset {\bf C}^5$.
The Gr\"obner basis is complicated, but you'll find what you need
if you look hard enough!) 
\bigskip
\noindent
II.  (Envelopes)  A family of curves in the plane is given 
by an equation $F(x,y,t) = 0$.  Setting $t$ equal to a constant
gives one of the curves in the family.  In this problem, we will
study the family defined by the equation:
$$F(x,y,t) = (x - t)^2 + (y - t^2)^2 - 1 = 0$$
From the form, we see that each curve
$F(x,y,t_0)$ is a circle of radius 1 whose center is a point
on the parabola $(t,t^2)$.  We can visualize the curves in
such a family in Maple by generating an {\it animated} plot
where $t$ is the ``animation variable''.  It is most convenient
to do this in parametric form though, so we will consider the 
parametric equations:
$$\eqalign{x &= t + \cos(\theta)\cr
           y &= t^2 + \sin(\theta)\cr}$$
where the $(\cos(\theta),\sin(\theta))$ parametrizes the 
circle, and the $(t,t^2)$ give the coordinates of the center.
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\item{A)}  Load the {\tt plots} package:
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\centerline{\tt with(plots):}
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\item{} Then enter the following command:
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\centerline{\tt animate([t+cos(theta),t\^{}2+sin(theta),theta=0..2*Pi],t=-2..2,frames=20);}
\medskip
\item{}
To ``play the animation'', you first left click the mouse over the
graph, and select the options (forward or backward, loop or once, etc.) from 
the toolbar.  Then right click the mouse over the graph and select
play from the animate option on the plot menu.  

\bigskip
\item{}  The {\it envelope} of the family of curves given by $F(x,y,t) = 0$
is another curve that is {\it tangent} to each of the curves in the 
family at the intersection points of the envelope and the family curve.  
It is found by eliminating $t$ between the two equations:
$$\eqalign{F(x,y,t) &= 0\cr
           {\partial\over\partial t} F(x,y,t) &= 0\cr}$$
(We will discuss this in more detail in class next week.)
For now, let's take this as given and find the envelope for
the family of curves above. 

\bigskip
\item{B)}  Define $F = (x-t)^2 + (y-t^2)^2 - 1$ as above, and compute
an appropriate Gr\"obner basis to find generator(s) of the elimination
ideal
$$I_1 = \langle F(x,y,t), {\partial\over\partial t}F(x,y,t) \rangle \cap {\bf C}[x,y].$$
\item{C)}  Use {\tt implicitplot} to generate a picture of the envelope
curve for this family of circles (the variety $V(I_1)$ with $I_1$ as in part B).  
\item{D)}  To understand the geometry here in more detail, it will help
to generate a composite plot showing the animated family of curves,
together with the implicit plot of the envelope, and 
also the path followed by the centers of the circles in the family.  
The {\tt display} command in the {\tt plots package} can be used to 
combine plots in this way.  Look up the online help and see if you 
can figure out how it should work.  
\bigskip
\noindent
{\it Additional Problems}
\bigskip
\noindent
III.  The {\it Whitney umbrella} is the surface in ${\bf R}^3$
given parametrically by 
$$\eqalign{x &= uv\cr
           y &= v\cr
           z &= u^2\cr}$$
\item{A)}  Find the equation of the smallest variety containing
the Whitney umbrella.
\item{B)}  Show that the parametrization fills up the variety over
${\bf C}$ but not over ${\bf R}$.  Over ${\bf R}$, exactly what
points are omitted?
\item{C)}  Show that the parameters $u,v$ giving a point 
on the Whitney umbrella are not always 
uniquely determined by $x,y,z$ (that is, the parametrization
is not one-to-one).  Find the points where uniqueness
fails, and explain your answer using a plot of the surface (the
parametric plot will be most useful here).  
\bigskip
\noindent
IV.  Prove Theorem 2 in Section 2 of Chapter 3 (the ``geometric
Extension Theorem'') using Lemma 1 and the Extension Theorem from
Section 1.
\bigskip
\noindent
V.  You have just discovered the remnants of an ancient alien
civilization in a subterranean complex.  Near the entrance
there are two large rooms connected by a short corridor 3 feet
wide and 4 feet long.  The floor plan is given in the diagram 
below:
\vglue 3.0truein
In Room 1, at the point marked A, you have discovered a functional, 
but very fragile, robot.  It has a horizontal cross-section that is a 
circle of diameter 2 feet.  The robot will accept movement commands,
but will not let itself be moved by any other means.  Your job 
is to figure out a way (any way) to maneuver the robot from point 
A to point B, subject to the following constraints:
\medskip
\item{1)}  The aliens were apparently very clever mathematicians,
but the robot's control program is {\it very limited} -- the
way you must specify its motion is to give a single polynomial
parametric curve $x = f(t), y = g(t)$ that traces the path
of the {\it center} of the robot's circular cross-section.
Only one curve is allowed -- no piecewise polynomial paths.
\item{2)}  The robot will be damaged beyond repair if it runs
into one of the walls.  If it does, you will have destroyed
the most amazing archeological discovery of all time.  
{\it So be careful!}
\item{3)}  Fortunately, you get to ``simulate'' possible 
motions using Maple as often as you like until you find
one that works, and then you can give the information
to the robot.
\medskip
\noindent
For this problem you will need to generate a Maple worksheet
showing the rooms and corridor, and a display showing
that your path will keep the robot from hitting the walls
at any point.  And, {\it give a justification for why your path
will not run the robot into the walls}.  (By this I mean more evidence
than the picture -- for instance, if feasible, you could find the point on 
the envelope that is closest to the walls, and show that the
distance at the closest point is strictly positive.)
\bigskip
\noindent
{\it Suggestions}
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\item{1)}  A good first step will be to set up a convenient
coordinate system.  This will let you set up plots showing the
rooms, etc.  You can also experiment and determine some 
paths from point A to point B to try.
\item{2)}  The discussion of envelopes from question II above
is relevant here!
\item{3)}  {\it Important Note:}  You will need to be somewhat clever
in how you apply the techniques we have discussed -- if you use
a ``brute force'' method, to compute the envelope for some
``obvious'' choices of the path, then you can easily end
up giving Maple a Gr\"obner basis calculation that will use
up an hour of calculation and tie up the network server
without producing any results(!)  There are several things
you can experiment with:
\medskip
\item{$\bullet$} the monomial order you use (look at the online
help for MonomialOrder)
\item{$\bullet$}  you can reduce the number of variables by
looking at one ``slice'' $x = const$ or $y = const$ at a
time.  This generally speeds things up {\it a lot}, but
you will need to look at several of these computations
to make sure the robot follows an acceptable path.
\bigskip
\item{}  However you approach the problem, you will need 
{\it give a justification for why your path
will not run the robot into the walls}.
\bye

