Mathematics 136, section 4 -- AP Calculus

Lab Day 2: Parametric Curves and Motion

September 23, 2003

Background

Today's lab is an extended demonstration to introduce a new idea via Maple's graphics facilities. The worksheet you develop today will be your class notes for the day, so make sure both partners in your lab pair print out a copy to keep. In class so far, we have looked only at graphs with equations y = f(x). These are familiar and easy to deal with, but they have certain limitations if we want to describe some situations like the path followed by an object moving in the plane. An object can start at one point and return to that point at some later time, its path can cross vertical lines more than once, etc. A graph y = f(x) cannot do either of those things. Hence to describe this kind of motion, we need a more general idea of how to describe curves in the plane. If we think of the position of the moving object as a function of time, then it is natural to think of describing the path by giving two different functions:

x = f(t); y = g(t)
giving the x- and y-coordinates of the object at each time. If we then plot the points (f(t),g(t)) for all t in the time interval over which the motion takes place, we will see a curve in the plane -- the path the object follows. The resulting curve is called a parametric curve: the time t is the ``parameter''. If you are used to thinking only of graphs y=f(x), though, parametric curves can ``take some getting used to'', because the values of t will not be apparent in the x-y plot of the path of motion. In other words, when we plot a parametric curve, we will see the path traced out over a certain time interval, but we will not directly see where we are at each time. To help visualize how all this works, then, we will look at some examples today using Maple's animation feature. This will let us see how parametric curves are ``traced out'' as t ranges over its domain. You will not need to generate the Maple commands for these animated plots yourself; I will provide those. The goal of today's lab is just to develop an understanding of how this new idea for describing curves works.

Lab Activities

Before proceeding to the demonstrations below, execute the following command in Maple:

with(plots):
(This loads in a package of graphics routines including the animation routine we will use today. (Note: the colon at the end means the output -- the list of names of the procedures in the plots package -- will not be printed. If you want to see all the stuff that is in there, replace the colon by the usual semicolon.)

A) Here is a first example of a parametric curve: x = 2cos(t); y = sin(t).
1. To see what this looks like for t starting at zero and going to 2 Pi, enter and execute the following plotting command:
  
  plot([2*cos(t),sin(t),t=0..2*Pi],scaling=CONSTRAINED);
  (This is the Maple format for all ``straight'' parametric plotting. In the plot command, inside [ ], you put the x-coordinate function, the y-coordinate function, then the range of t-values to be plotted, separated by commas.) What kind of curve is this? (Answer in a text region.)
2. A first way to visualize what is happening is to plot the x- and y-coordinate functions as functions of t. Do this and try to understand the correspondence between the parametric plot and these two plots. For instance, where does the curve start at t = 0? Where is it at t = Pi/2, Pi, 3*Pi/2, 2*Pi? Which direction are we moving around the curve as t increases? (Answer in a text region.)
3. Now, let's see if we were correct. The following command generates an animated plot, showing successively larger parts of the curve, so we can see it ``grow'' until the full range of t-values 0..2*Pi is attained:
  
  animate([2*cos(2*Pi*s*t),sin(2*Pi*s*t),t=0..1],s=0..1,frames=50,scaling=CONSTRAINED);
  When you press RETURN on this command, after a moment you will see a blank set of axes. Click the left mouse button on the plot region. If you look carefully at the toolbar, you will see that a new set of control buttons has appeared. These work the same way the control buttons on a cassette tape player work. To ``play'' the animation, press the button with the solid triangle pointing right. Experiment with the other animation control buttons to see what they do. In particular, you should see that you can speed up or slow down the animation, play continuously, reverse, etc. Describe what the animation shows you in a text region.
B) Now a second example: x = t² -1; y = t³ - t. We'll look at the part of this one for -2 <= t <= 2.
1. First plot the parametric curve:
  
  plot([t^2-1,t^3-t,t=-2..2],scaling=CONSTRAINED);
  Also plot the x- and y-coordinate functions and answer these questions: What points on the curve correspond to t = -2, -1, 0, 1, 2? Which direction are we moving along the curve as t increases?
2. Now enter:
  
  animate([(-2+4*s*t)^2-1,(-2+4*s*t)^3-(-2+4*s*t),t=0..1],s=0..1,
  frames=50,scaling=CONSTRAINED);
  (Maple will ``wrap'' the input onto a second line automatically; don't be concerned that this won't all fit on one line) and ``play'' the animation. Does this agree with your answers to part 1?
C) A final example: Parametric curves are especially useful for thinking about ``complicated'' motions like the path of a moon which is rotating about its planet, which is in turn rotating about a sun, which is in turn rotating about the center of its galaxy, ... To simplify things, let's do things relative to the sun, which we will consider as fixed (i.e. not moving) at (0,0). Then a possible type of orbit for the moon would be described by parametric equations like this: x = A cos(2 Pi t/P) + a cos(2 Pi t/p), and y = B sin(2 Pi t/P) + b sin(2 Pi t/p) where A,B,a,b,P,p are various constants. (This is the ``superposition'' of two different elliptical orbits -- the orbit of the planet about the sun, and the orbit of the moon around the planet, if you consider the planet as fixed.)
1. Here's a particular example:
  
  plot([10*cos(t)+(0.9)*cos(20*t),9*sin(t)+sin(20*t),t=0..2*pi]);
  Plot and describe.
2. Now let's animate this to see what the moon would do over time:
  
  animate([10*cos(s*t)+(.9)*cos(20*s*t),9*sin(s*t)+sin(20*s*t), t=0..2*Pi],s=0..1,
  scaling=CONSTRAINED,numpoints=200,frames=100);
  Note: This one will take a bit longer because we are plotting more points on each curve and more animation ``frames''. Describe. It will also be fun to change some of the constants here and observe the different curves you obtain.

Lab Assignment

Prepare a Maple worksheet showing all your computations and graphs for questions A,B,C above. Due: Wednesday, September 24.