Math 2325 
Project 3 (Chapter 9) - Parametric Curve Representation

Helmut Knaust, Department of Mathematical Sciences, UTEP, El Paso TX 79968
hknaust@utep.edu
10/7/2012. Modified from a version written by Art Duval.Last edit 10/3/2021

Proof Without Words

This is for the case p=3 q=1, r=2, s=4.

Rather than graphing (x(t), y(t)) over the entire interval [0,2π], the following animation graphs two portions of the graph simultaneously.
The first part graphed  in green is of (x(t), y(t)) for t in [0,π], while the second part in blue is of (x(π+t),y(π+t)) for t in [0,π].

Other cases can be handled in a somewhat analogous fashion.

{p,q,r,s}={3,1,2,4};
Clear[x,y,t,starttime, endtime,win];
x[t_]=Sin[p t]+Cos[q t];
y[t_]=Sin[r t]+Cos[s t];
starttime=0;
endtime=π;
win=2;
(*Do not change anything below this line*)
Animate[Show[
ParametricPlot[{x[u],y[u]},{u,π+starttime,π+t}, ImageSize->400,PlotRange -> {{-win,win},{-win,win}},PerformanceGoal->"Quality",
PlotStyle->Directive[Thick,Blue]],Graphics[{Red,Thickness[.01],Circle[{x[π+t],y[π+t]},.02]}],
ParametricPlot[{x[u],y[u]},{u,starttime,t}, PlotRange -> {{-win,win},{-win,win}},PerformanceGoal->"Quality",
PlotStyle->Directive[Thick,Green]],Graphics[{Red,Thickness[.01],Circle[{x[t],y[t]},.02]}],
Graphics[Text[StringJoin["t = ",ToString[NumberForm[t/Pi,{2,1}]],ToString[Pi,StandardForm]],{.75win,.95win}]]],
{t,starttime+(endtime-starttime)/10.^5,endtime},AnimationRepetitions->1,AppearanceElements->"ResetButton",ControlPlacement->Bottom]

The animation illustrates that for all t in [0,π]
	x(t+π) = - x(t)
	y(t+π) =   y(t) 
Can you prove this using trigonometric identities?