Mathematics 162                              Laboratory 11                        Week of April 12, 1993
	  
Name: _____________________________   Lab Partner: ___________________________
	  
Consulted with:  ____________________________________________________________

© Lafayette College, 1994.

Parametric Equations

You are familiar with curves described by equations in two variables.  For example, the graph of the equation x2 + y2 = 4 is the circle of radius 2 which is centered at the origin, and the graph of x = y2 is a parabola which is symmetric about the x-axis and opens to the right.  Furthermore, if f is a function of x, the graph of y = f (x) is a curve which has the property that no vertical line intersects the curve in more than one place.  The cycloid which you saw in the demonstration is the graph of a function, but it is not at all clear what a formula for that function might be.  The epicycloid is clearly not the graph of a function and it is very hard to guess what sort of equation might describe that curve.

Today, we are going to experiment with a new method for describing curves in the plane.  Let f (t) and g(t) be functions of the parameter t.  Let C be the curve consisting of all ordered pairs (f (t), g(t)), where a ≤ t ≤ b.  Then C is referred to as the curve with parametric equations

Part 1: Plotting Parameterized Curves

Consider the curve C given parametrically by

To plot this curve, we begin by defining the functions f  and g in Mathematica:

f[t_] := t Cos[t];
g[t_] := Sqrt[t] Sin[t]

If we were to attempt to plot the curve C by hand, we might begin by making a table of the ordered pairs (f (t), g(t)) for several values of t, like the one the following command generates:

table = Table[{f[t],g[t]}, {t,0,8,0.2}];
% //TableForm

Then we would plot the points in that table.

plot1 = ListPlot[table, PlotStyle->Red]

Then we would probably connect the dots in order with a smooth curve.

plot2 = ListPlot[table, PlotStyle->Blue,
              PlotJoined->True]

(Mathematica connects the dots with line segments, which is why it needs so many points.)  Mathematica has a single command, ParametricPlot, which will do all of this for us.

plot3 = ParametricPlot[{f[t], g[t]}, {t,0,16},
               PlotStyle->Green]

Use the command below to see that ParametricPlot does what it should.

Show[plot1,plot2,plot3]

Now use the same parametric equations in ParametricPlot, but change the interval for the parameter to 0 ≤ t ≤ 16.  How does the new curve compare with the original curve?

Consider now the parametric equations

Use ParametricPlot to plot the curve parametrized by this equation for 0 ≤ t ≤ π.  Add the option AspectRatio->Automatic to the command to eliminate distortion in the plot.  Compare this with the curves generated over the intervals 0 ≤ t ≤ 2π, and 0 ≤ t ≤ 4π.  (Plot these as well.)  How are these different from each other?  In particular, how are the last two curves plotted different from one another?

What is the difference between these curves and those that are parametrized by

(Hint:  Plot these equations for 0 ≤ t ≤ π and 0 ≤ t ≤ 2π.)

Rewrite the parametric equation (1) above to eliminate the parameter.  That is, find an equation that uses only x and y to describe the curve.

In the same way, rewrite equation (2) to eliminate the parameter.

Explain briefly (two or three sentences) how parametric curves give you more flexibility in describing curves than traditional xy equations.

Part 2: An Application

A simple application of parametric equations is describing the position of a moving object.  If a projectile is fired from ground level with initial velocity v0 at an angle of inclination Θ, then its position after t seconds of flight is

(Here we are ignoring air resistance.)  Define these as our new f and g:

f[t_] := v0 Cos[theta] t;
g[t_] := -16 t^2 + v0 Sin[theta] t

and choose particular values for v0 and Θ:

v0    = 800;
theta = Pi/6;

Now we can plot the trajectory of the projectile for 15 seconds, say, with the command

ParametricPlot[{f[t], g[t]}, {t,0,15}]

For each of the angles Θ = π/6, Θ = π/4, and Θ = π/3, plot the trajectory of the projectile until it returns to the ground, and display all three trajectories on one plot.  Print and label a copy and hand it in.  For which of these values of Θ does the projectile travel furthest horizontally?  (Hint:  The projectile hits ground when g(t) = 0 at some t > 0.  Find the appropriate t value—it will be a function of v0 and Θ—and make it the end of the t interval in your plot.)  For what Θ does it attain the highest altitude?

Part 3:  Some pretty pictures

There is nothing for you to write in this part of the lab and you are not absolutely required to work through it.  However, plotting the curves described below may help you to get a better feeling for curves described parametrically and will certainly show you some interesting pictures.

Curves (a)-(c) are known as Lissajous figures.

The curve in (d) is a cycloid like the one in the first animation at the start of lab.  Can you figure out how to modify this command to generate the curve shown in the second animation? (Bonus points for this!)

For a fixed r, this curve is called an epicycloid.  Try several values of r: r = 1 is simplest; r = 1/3 is more interesting, and r = 1/√2 is VERY complex.  For the last case, to fully appreciate the complexity, use the interval 0 ≤ t ≤ 40π, and increasethe number of t values where Mathematica computes points on the curve by adding the option PlotPoints->24.  This will slow down your computer quite a bit.