Math 2325 (Chapter 9) -- Parametric Curve Representation

General Plots

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,PlotStyle->Directive[Thick,Black]](*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,PlotStyle->Directive[Thick,Black]]

Shortcut:Param2

Param2, below, 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.   Note that you must define it before running it!

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,PlotStyle->Directive[Thick,Black]];

Param2[1,2,1,1]

Animation

The following command will animate a parametric plot.  You can modify: the x-function; the y-function; the interval for the parameter; and the range of the graphing window.

Clear[x,y,t,starttime, endtime,win];
x[t_]=Cos[t];
y[t_]=Sin[t];
starttime=0;
endtime=2*Pi;
win=2;

(*Do not change anything below this line*)
Animate[Show[
ParametricPlot[{x[u],y[u]},{u,starttime,endtime}, PlotRange -> {{-win,win},{-win,win}}],
ParametricPlot[{x[u],y[u]},{u,starttime,t}, PlotRange -> {{-win,win},{-win,win}},
PlotStyle->Directive[Thick,Black]],Graphics[{Red,Thickness[.01],Circle[{x[t],y[t]},.02]}],
Graphics[Text[StringJoin["t=",ToString[t/Pi],ToString[Pi,StandardForm]],{.75win,.95win}]]],
{t,starttime,endtime}]

To change the size of the graphing window, pause the animation, click on the graph (until you get a yellow box), grab a corner, and resize it.

Art Duval, artduval@math.utep.edu
February 13, 2005;
animation updated to Mathematica 6
September 22, 2008;
minor improvements
February 16, 2009;
animation originally inspired by 
Tamara Olson, Michigan Technological University, 2006