Manipulate[ParametricPlot[{{(1+Cos[f])Cos[f-t],(1+Cos[f])Sin[f-t]+t+Sin[t]},{1+Cos[f],f+Sin[f]},{0.1 Cos[f],0.1 Sin[f]+t+Sin[t]}},{f,0,4Pi},PlotRange->{{-2,2},{-2,14}}],{t,0,10}]

Manipulate[ParametricPlot[{{(1.5+Cos[f])Sin[f-t]+1.5t+Sin[t],(1.5+Cos[f])Cos[f-t]},{1.5f+Sin[f],1.5+Cos[f]},{0.1 Cos[f]+1.5t+Sin[t],0.1 Sin[f]}},{f,0,10Pi},PlotRange->{{-2,30},{-3,4}}],{t,0,30}]

Manipulate[ParametricPlot[{{(1.2+Cos[3 f])Cos[f-t],(1.2+Cos[3 f])Sin[f-t]+1.2t+1/3Sin[3t]},{1.2+Cos[3f],1.2f+1/3Sin[3f]},{0.1 Cos[f],0.1 Sin[f]+1.2t+1/3Sin[3t]}},{f,0,3Pi},PlotRange->{{-2,3},{-2,9}}],{t,0,10}]

Manipulate[ParametricPlot[{{(1-Cos[f]Sin[3 f])Cos[f-t],(1-Cos[f]Sin[3 f])Sin[f-t]+t-(1/2) (2+Cos[2 t]) ((Sin[t])^2)},{(1-Cos[f]Sin[3 f]),f-(1/2) (2+Cos[2 f]) ((Sin[f])^2)},{0.1 Cos[f],0.1 Sin[f]+t-(1/2) (2+Cos[2 t]) ((Sin[t])^2)}},{f,0,3Pi},PlotRange->{{-2,3},{-2,9}}],{t,0,10}]

Manipulate[ParametricPlot[{{(2Cos[ f])Cos[f-t],(2Cos[ f])Sin[f-t]+ 2 Sin[t]},{2Cos[ f],2Sin[f]},{0.1 Cos[f],0.1 Sin[f]+2 Sin[t]}},{f,0,2Pi},PlotRange->{{-3,3},{-3,3}}],{t,0,20}]

Manipulate[ParametricPlot[{{(2+(∑_(n=1)^100)((Cos[n n f])/( n n )))Cos[f-t],(2+(∑_(n=1)^100)((Cos[n n f])/( n n )))Sin[f-t] + 2 t+(∑_(n=1)^100)((Sin[n n t])/( n n n n ))},{2+(∑_(n=1)^100)((Cos[n n f])/( n n )),2 f+(∑_(n=1)^100)((Sin[n n f])/( n n n n ))},{0.1 Cos[f],0.1 Sin[f]+ 2 t+(∑_(n=1)^100)((Sin[n n t])/( n n n n ))}},{f,0,3Pi},PlotRange->{{-4,4},{-1,14}}],{t,0,10}]

Manipulate[ParametricPlot[{{(3+(∑_(n=1)^15)((Cos[3^n f])/(2^n)))Cos[f-t],(3+(∑_(n=1)^15)((Cos[3^n f])/(2^n)))Sin[f-t] + 3 t+(∑_(n=1)^15)((Sin[3^n t])/( 2^n 3^n))},{3+(∑_(n=1)^15)((Cos[3^n f])/(2^n)), 3 f+(∑_(n=1)^15)((Sin[3^n f])/( 2^n 3^n))},{0.1 Cos[f],0.1 Sin[f]+  3 t+(∑_(n=1)^15)((Sin[3^n t])/( 2^n 3^n))}},{f,0,2Pi},PlotRange->{{-4,4},{-1,14}}],{t,0,10}]

Manipulate[ParametricPlot[{{(Exp[f])Cos[f-t],(Exp[f])Sin[f-t]+Exp[t]},{(Exp[f]),Exp[f]},{0.3 Cos[f],0.3 Sin[f]+Exp[t]}},{f,-5,3},PlotRange->{{-10,20},{-2,20}}],{t,-2,10}]

Manipulate[ParametricPlot[{{(E^(-y y)) Cos[(1/2) (Sqrt[π]) Erfi[y]-(1/2) (Sqrt[π]) Erfi[t]],(E^(-y y)) Sin[(1/2) (Sqrt[π]) Erfi[y]-(1/2) (Sqrt[π]) Erfi[t]]+t},{(E^(-y y)),y},{.05 Cos[5y],.05Sin[5y]+t}},{y,0,3},PlotRange->{{-1,1},{-.2,3}}],{t,0,3}]

f[y_]:=N[3.165+ (∑_(n=1)^20)((((-1))^(n-1))/(((2 n-1))^2))Sin[(2 n-1)y]];
φ[y_]:=NIntegrate[(1/(f[t])),{t,0,y}];
ParametricPlot[{{f[y] Cos[φ[y]],f[y] Sin[φ[y]]},{f[y],y},{.15 Cos[5y],.15Sin[5y]}},{y,-5,15}]