Math 2325
(Chapter 9) - Parametric Curve Representation

Note: Be sure to run the following command first, before anything else!

<< Graphics`Animation`

General parametric plots

Static

The following shows how to draw general parametric plots, with comments briefly explaining each piece of the command.

ParametricPlot[
{Cos[t], 
        Sin[t]},                                         (*x - function,  y - function*) 
{t, 0, 
        2*Pi},                                                  (*parameter and its range*) 
PlotRange  -> {{-2, 2}, {-2, 
            2}},          (*limits of graphing window*) 
AspectRatio -> 1, 
      PlotPoints -> 1000] (*technical stuff*)

Here's the same command, without the comments, to make it easier for you to modify and draw your own parametric plots.

ParametricPlot[ {Cos[t], Sin[t]}, {t, 0, 2*Pi}, 
      PlotRange  -> {{-2, 2}, {-2, 2}},  
      AspectRatio -> 1, PlotPoints -> 1000]

Animated

The following shows how to "animate" parametric plots, again with comments briefly explaining each piece of the command.  You must use "u*t" for the parameter (unless you want to do some further monkeying).  The result will be a series of pictures or "frames" that, if you look at in sequence, will appear to draw the curve.  In particular, if you set the height of the window of the computer just right, repeatedly hitting "PageUp" will produce a crude movie.

\(\(\(MovieParametricPlot[
{Cos[u*t], 
        Sin[u*t]}, \  (*x - function, \ 
        y - function*) 
 (*NOTE : \ 
          You\ must\ use\ "u*t"\ for\ the\ variable\ \(here!\)*) 
{t, 0, 
        2  Pi}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (*parameter\ and\ its\ range*) 
{u, 0, 
        1}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (*technical\ stuff*) 
Frames -> 
        24, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (*number\ of\ frames\ in\ "movie"*) 
PlotRange\  -> \ {{\(-2\), 2}, {\(-2\), 
            2}}, \ \  (*limits\ of\ graphing\ window*) 
AspectRatio -> 
        Automatic,  (*technical\ stuff*) 
PlotPoints -> 1000]\)\( (*more\ technical\ stuff*) \)\)\)

Once again, here is the same command, without the comments, to make it easier for you to modify and make your own movies.

MovieParametricPlot[{Cos[u*t], Sin[u*t]},  {t, 0, 2  Pi}, {u, 0, 1}, 
      Frames -> 24,  
      PlotRange  ->  {{-2, 2}, {-2, 2}},  
      AspectRatio -> Automatic,  PlotPoints -> 1000]

Param2 program

The following functions will make it easy for us to investigate the curves described in Section 9.2.3, namely, the parametric curves x=sin(pt)+cos(qt), y=sin(rt)+cos(st) over the interval 0 ≤ t ≤ 2π.  Param2 just draws the curve; AniParam2 makes the movie that shows how the curve is drawn.  Note that you must define the commands before running them!

Param2[p_, q_, r_, s_] := 
        ParametricPlot[{Sin[p*t] + Cos[q*t], Sin[r*t] + Cos[s*t]}, {t, 0, 
            2*Pi}, PlotRange -> {{-2, 2}, {-2, 2}}, 
          AspectRatio -> 1,  
          PlotPoints  -> 1000];
AniParam2[p_, q_, r_, s_] := 
        MovieParametricPlot[{Sin[p*u*t] + Cos[q*u*t], 
            Sin[r*u*t] + Cos[s*u*t]}, {t, 0, 2*Pi}, {u, 0, 1}, 
          Frames -> 24, AspectRatio -> Automatic, 
          PlotRange -> {{-2, 2}, {-2, 2}}, 
          PlotPoints  -> 1000];

Param2[1, 2, 1, 1]

AniParam2[1, 2, 1, 1]

Art Duval, artduval@math.utep.edu
February 13, 2005