Manipulate[
ParametricPlot[{t*Cos[t],t*Sin[t]},{t,0,tmax},PlotStyle->Thick,PlotRange->{{-10π,10π},{-10π,10π}}],
{{tmax,0.01},0.01,10π}]

(*Cycloid*)

Manipulate[
ParametricPlot[{t-Sin[t],1-Cos[t]},{t,0,tmax},PlotStyle->Thick,PlotRange->{{0,10π},{0,2}}],
{{tmax,0.01},0.01,10π}]

Manipulate[With[{t=2Pi r},Show[{ParametricPlot[#[s],{s,10^-9,t},AxesOrigin->{0,0},PlotRange->{{-1,4Pi+1.1},{-1 - 0.01,3 + 0.01}},Ticks->None,ImageSize->450],Graphics[{Red,Circle[{t,1}],Black,Point[{t,1}],Line[{{t,1},#[t]}],Point[#[t]]}]}]&[{(#-ratio Sin[#]),(1-ratio Cos[#])}&]],{{r,0,"revolutions"},0,2,Appearance->"Labeled"},{{ratio,1,"ratio"},0,2,Appearance->"Labeled"}]

(*Spirograph*)

Manipulate[
ParametricPlot[{(8/7)*Cos[t]+0.5*Cos[(8/7)*t],(8/7)*Sin[t]-0.5*Sin[(8/7)*t]},{t,0,tmax},PlotStyle->Thick,PlotRange->{{-2,2},{-2,2}}],
{{tmax,0.01},0.01,14π}]

Manipulate[
ParametricPlot[{Sin[t]*(Exp[Cos[t]]-2*Cos[4*t]-Sin[((1/12)t)^5]),Cos[t]*(Exp[Cos[t]]-2*Cos[4*t]-Sin[((1/12)t)^5])},{t,0,tmax},PlotPoints->1000,PlotRange->{{-4,4},{-4,4}}],
{{tmax,0.01},0.01,6π}]