Animate[Show[ParametricPlot[{t-Sin[t],1-Cos[t]},{t,0,x},PlotRange->{{-1.1,21},{-0.1,2.1}},PlotStyle->Red],ParametricPlot[{x+Cos[t],1+Sin[t]},{t,0,2Pi}, PlotRange->{{-1.1,21},{-1.1,2.1}},PlotStyle->Black], Graphics[{Red,PointSize[Medium],Point[{x-Sin[x],1-Cos[x]}]}]],{x,0,6Pi}]

Animate[Show[ParametricPlot[{t-3Sin[t],1-3Cos[t]},{t,0,x},PlotRange->{{-2.1,22},{-2.1,4.1}},PlotStyle->Red],ParametricPlot[{x+Cos[t],1+Sin[t]},{t,0,2Pi}, PlotRange->{{-2.1,22},{-2.1,4.1}},PlotStyle->Black], ParametricPlot[{x+3Cos[t],1+3Sin[t]},{t,0,2Pi}, PlotRange->{{-2.1,22},{-2.1,4.1}},PlotStyle->Green],Graphics[{Red,PointSize[Medium],Point[{x-3Sin[x],1-3Cos[x]}]}]],{x,0,6Pi}]

Animate[Show[ParametricPlot[{2t-Sin[t],2-Cos[t]},{t,0,x},PlotRange->{{-2.1,42},{-2.1,4.1}},PlotStyle->Red],ParametricPlot[{2x+2Cos[t],2+2Sin[t]},{t,0,2Pi}, PlotRange->{{-2.1,42},{-2.1,4.1}},PlotStyle->Black],
ParametricPlot[{2x+Cos[t],2+Sin[t]},{t,0,2Pi}, PlotRange->{{-2.1,42},{-2.1,4.1}},PlotStyle->Green], Graphics[{Red,PointSize[Medium],Point[{2x-Sin[x],2-Cos[x]}]}]],{x,0,6Pi}]