Curves Defined By Parametric Equations.

Section 10.1 - Stewart 7th ed.

Figure 7

r=0.2;

x1[t_]=t+2Sin[2t];
y1[t_]=t+2Cos[5t];

PATH1=ParametricPlot[{x1[t],y1[t]},{t,-2π,2π},PlotRange->{{-6.5,6.5},{-8,8}}];

Manipulate[Show[{PATH1,Graphics[Disk[{x1[t],y1[t]},r]]},AspectRatio->1],{t,-2π,2π,(π/50)}]

Figure 12

r=0.05;

x2[t_]=Sin[t]+(1/2)Cos[5t]+(1/4)Sin[13t];
y2[t_]=Cos[t]+(1/2)Sin[5t]+(1/4)Cos[13t];

PATH2=ParametricPlot[{x2[t],y2[t]},{t,-2π,2π},PlotRange->{{-1.5,1.5},{-3,3}},PlotPoints->100];

PATH3[T_]:=ParametricPlot[{x2[t],y2[t]},{t,0,T+0.00001},PlotRange->{{-1.5,1.5},{-3,3}},PlotPoints->100];

Manipulate[Show[{PATH3[t],Graphics[Disk[{x2[t],y2[t]},r]]},AspectRatio->1],{t,0,2π,(π/1000)}]

Example 7: The Cycloid

r=.1;
R=1;

x3[t_]=R(t-Sin[t]);
y3[t_]=R(1-Cos[t]);

PATH4[T_]:=ParametricPlot[{x3[t],y3[t]},{t,0,T+0.00001},PlotRange->{{0,6π},{-1,5}}];

Manipulate[Show[{PATH4[t],Graphics[{Disk[{x3[t],y3[t]},r],Circle[{R t,R},{R,R}],Line[{{R t,R},{x3[t],y3[t]}}]}]},AspectRatio->.31],{t,0,6π,(π/10)}]

Example 8: Conchoids of Nicomedes

Manipulate[
Show[{
ParametricPlot[{a+Cos[t],a Tan[t]+Sin[t]},{t,-π,((-π)/2)-.0001},PlotRange->{{-4,4},{-4,4}}],
ParametricPlot[{a+Cos[t],a Tan[t]+Sin[t]},{t,((-π)/2)+.0001,(π/2)-.0001},PlotRange->{{-4,4},{-4,4}}],
ParametricPlot[{a+Cos[t],a Tan[t]+Sin[t]},{t,(π/2)+.0001,π},PlotRange->{{-4,4},{-4,4}}]
}],
{a,-2,2,0.1}]