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\centerline{Mathematics 131 -- Calculus for Physical and Life Sciences 1}
\centerline{Lab Day 3 -- Parametric Curves}
\centerline{November 1, 2004}
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{\it Background}
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A parametrized curve in the plane
gives the position of a particle or moving point as a function of
time.  For this we need to know both the $x$ and $y$
coordinates as functions of time:
$$\cases{x = x(t) & \cr
         y = y(t) & \cr}$$
The form and shape of the curve ``swept out'' by the particle depends 
on the specific
functions $x(t)$ and $y(t)$. In 
this lab you will use Maple to plot several parametrized
curves that are not immediate to 
visualize, and/or difficult to draw by hand. 
In your lab report worksheet, hand in plots of all these parametric 
curves, and answer the questions in text regions.

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{\it Lab Work}

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A)  Start by entering the command
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\centerline{\tt plot([cos(t),sin(t),t=0..2*Pi],scaling=constrained);}
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to plot the parametric curve with coordinate functions 
$x(t) = \cos(t)$ and $y(t) = \sin(t)$.
(Note the square brackets around the
component functions and time interval.  The
``scaling'' option forces Maple to use the same horizontal and
vertical scales; {\it always include this today}.)  What curve is this?  Why?  (Hint:
What is true about $(x(t))^2 + (y(t))^2$ for all $t$?)
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B)  Plot the parametric curve with coordinate functions
$x(t)=2t/(t^2+1)$, $y(t)=(t^2-1)/(t^2+1)$ for $-100\leq
t\leq 100$. Identify this curve visually, and use algebra to show
that your guess is correct.
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C)  Next, plot the parametric curve with component functions
$x(t)=t-\sin t$, $y(t)=1-\cos t$ for $0\leq t\leq 6\pi$. 
(As above $\pi$ is {\tt Pi} in Maple.)
Find the values of $t$ at which the velocity is zero. 
(Velocity is zero when $x'(t) = 0$ and $y'(t) = 0$.)
Where on the curve are these points located?
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D)
The curve from question C is called a {\it cycloid}.
It is the path traced by a
  point on the rim of a rolling wheel of radius 1. By changing the
  multiplier on the trig functions, you can simulate motion on other
  wheels. For example, $x(t)=t-1.2\sin t$, $y(t)=1-1.2\cos t$ might
  represent the path of a point on the rim of a train wheel, while
  $x(t)=t-0.9\sin t$, $y(t)=1-0.9\cos t$ might be the path of a
  bicycle reflector. (Plot these curves as well to see the changes.)
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{\it Comment:}
  Unlike a function graph, a parametrized curve can even have
  cusps or corners at points where both coordinate functions are
  differentiable if the velocity is zero there. Intuitively, the point 
  can
  come to a stop momentarily, then proceed in a different direction, 
  thereby
  tracing a curve with no tangent line. 
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E) Plot $x(t)=\cos^3t$, $y(t)=\sin^3t$, $0\leq t\leq 2\pi$. Find all
  points on the curve where the velocity is zero.  Where on the curve
  are these points located? Use algebra to show that this curve is
  given implicitly by $x^{2/3}+y^{2/3}=1$.
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F) Plot $x(t)=e^{t/2\pi}\cos t$, $y(t)=e^{t/2\pi}\sin t$, $-4\leq
  t\leq4$. Try graphing $x(t)=e^{kt}\cos t$, $y(t)=e^{kt}\sin t$ for a
  few values of~$k$ (both positive and negative). What happens as
  $k\to 0$? How do the curves with $k>0$ differ from those with $k<0$?
  Give examples of two natural forms (living
  organisms from at least two biological kingdoms) that this
  curve resembles.

  Explanation: Curves in this family are known as {\it logarithmic
  spirals}, and have the property that they meet every radial line
  through the origin at a constant angle. This property explains their
  appearance in biological growth.
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G) Plot $x(t)=t\cos t$, $y(t)=t\sin t$, $0\leq t\leq 6\pi$.

  Explanation: This curve is a {\it spiral of Archimedes}, and these can be used
  to trisect angles together with a compass and straightedge. It is a
  famous theorem that a general angle cannot be trisected using {\bf only} a
  compass and straightedge.
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H) Plot $x(t)=\cos(3t)\cos t$, $y(t)=\cos(3t)\sin t$, for $0\leq t\leq
  2\pi$. How many ``petals'' do you think $x(t)=\cos(5t)\cos t$,
  $y(t)=\cos(5t)\sin t$, $0\leq t\leq 2\pi$, has? Plot the curve and
  see if your guess is correct.

  What about $x(t)=\cos(2t)\cos t$, $y(t)=\cos(2t)\sin t$, for $0\leq
  t\leq 2\pi$? Again, try it and see. If you can, explain the outcome
  of this experiment. How many petals do you think $x(t)=\cos(4t)\cos
  t$, $y(t)=\cos(4t)\sin t$ has?

  Explanation: These curves are often called ``rose curves'', for the
  obvious reason that they look like stylized flowers.
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I) 
Consider a circular ring of inner radius $96$, inside of which
  is rolled a wheel of radius $r < 96$.  A point on the edge of
   the wheel traces the
  ``hypocycloid'' parametrized by
$$\cases{
  x(t) = (96-r)\cos t + r\cos((96-r)t/r) & \cr
  y(t) = (96-r)\sin t - r\sin((96-r)t/r) & \cr
}$$
(The ``Spirograph'' toy you may have played with as a child draws
these curves.)
  Plotting over $0\leq t\leq 2\pi$ will usually not give the entire curve;
  use a large multiple like $0$ to $100\pi$ for the plot interval.
  Plot the resulting curves for $r=24$, $30$, $35$, $36$, $40$, $52$,
  $56$, $63$, $64$, $72$, and $84$.
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Note:  If you want to get a nice visual explanation for how
these curves are generated, go to a web browser and look at
the page {\tt http://mathworld.wolfram.com/Hypocycloid.html}
and scroll down the page until you see the animations.
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{\it Assignment}
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Lab writeups due in class on Monday, November 8.
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