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\centerline{Mathematics 36, section 5 -- AP Calculus}
\centerline{Lab 7 -- Parametric Curves}
\centerline{November 30, 1999}
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\noindent
{\it Goals}
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Today, we want to use Maple to plot parametric curves and get
more familiar with the concept by seeing a collection of 
interesting examples.
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\noindent
{\it Parametric Curve Plotting in Maple}
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The {\tt plot} command that we have used to draw graphs
$y = f(x)$ in the plane can also be used for parametric curve
plotting.  The format is somewhat different however:
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\centerline{\tt plot([x-coord,y-coord,t-range]);}
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\noindent
where {\tt x-coord} and {\tt y-coord} are the expressions for 
the $x$ and $y$ coordinates on the curve as functions of the
parameter $t$, and {\tt t-range} is the range of $t$ values 
that you want to see plotted.  For example, try the 
command:
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\centerline{\tt plot([3*cos(t)+1,3*sin(t),t=0..2*Pi]);}
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Maple's default in plotting is to try to ``fill up'' the 
screen as much as possible.  This means that different scales
are often used on the two coordinate axes.  Look carefully
at your plot and notice that this has happened.  To {\it
prevent this from happening} and get a true 1:1 ratio between
the $x$-axis and $y$-axis scales, you can add an option to
the plot command {\tt scaling=CONSTRAINED}.  Try this and
reexecute your plot command to see the effect. 

To plot more than one parametric curve together on the 
same set of axes, you can put the specifications for the
different curves into a set of curly brackets $\{, \}$, 
and the options outside.  For instance, try 
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\centerline{\tt plot($\{$[cos(t),sin(t),t=0..Pi],[2-3*t,2*t,t=0..1]$\}$,
scaling=CONSTRAINED);}
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Which curve is which?
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\noindent
{\it Lab Questions}
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\item{A)}  How would you use parametric curve plotting
to plot a circle of any given radius $r$ with center at 
any given point $(h,k)$?  Illustrate your method
by showing the circles:
\item{} center $(0,0)$, radius $r = 1$,
\item{} center $(0,0)$, radius $r = 3$,
\item{} center $(-1,2)$, radius $r = 4$
\item{} together in the same plot.
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\item{B)} An ellipse ${x^2\over a^2} + {y^2\over b^2} = 1$
can be thought of as a circle of radius 1 ``stretched or shrunk''
by a factor of $a$ in the $x$-direction, and by a factor
of $b$ in the $y$-direction.  Explain how this leads to 
parametric equations for ellipses.  Also explain how
to shift the whole ellipse so that its center is at
a general given point $(h,k)$ rather than at $(0,0)$.
Illustrate by showing ellipses:
\item{} center $(0,0)$,  $a = 1$, $b = 2$
\item{} center $(1,0)$,  $a = 3$, $b = 2$
\item{} center $(2,-1)$, $a = 4$, $b = 1$
\item{} together in the same plot.
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\item{C)}  Plot the parametric curve $x = t^2 - 1$, $y = t^3 - t$,
for $t \in [-2,2]$.  Why does the curve cross over itself 
at the origin (how many different $t$-values give $(x,y) = (0,0)$)?
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\item{D)}  {\it Lissajous figures}.  The Lissajous figures
are the parametric curves given by 
$$x = \cos(at) \qquad y = \sin(bt)$$
where the constants $a,b$ are any positive numbers.
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\itemitem{1)}  If $a,b$ are {\it integers}, then the
Lissajous figure forms a nice closed curve (it returns to
its starting point after some interval of $t$ values).
Plot the Lissajous figures with 
$$\cases{a = 2, b = 5 &\cr
         a = 2, b = 4 &\cr
         a = 3, b = 7 &\cr}$$
for $t \in [0,2\pi]$
(but not on the same axes -- that would be a real mess!)
What happens in the first one?  Why the two ``horns''?
(It may help to plot $x$ versus $t$ and $y$ versus $t$
separately to see what is going on here.)
Experiment with some different $a,b$, including the
case $a = 1, b=2$.  Is that one the same as one you 
saw before?
What is the general pattern of the shape of the Lissajous
figure with integer $a,b$?  Does it depend on whether
$a,b$ are odd or even?  Explain.
\itemitem{2)}  Now try the case $a = 2, b = \sqrt{3}$.
Take larger intervals $[0,L]$ for $L > 2\pi$.
What about $a = \pi, b = e$?
How are these cases different?

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\noindent
{\it Assignment}
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Individual lab reports, due Monday, December 6.
\bye
