Manipulate[Show[ParametricPlot3D[{x,2 +Sin[x],0},{x,0,2Pi},PlotStyle->{{Thickness[0.01],RGBColor[0,0,1/2]}}],ParametricPlot3D[{x,(2 +Sin[x])Cos[t],(2 +Sin[x])Sin[t]},{x,0,2Pi},PlotStyle->{{Thickness[0.01],RGBColor[1/2,0,1/2]}}],PlotRange->{{0,2Pi},{-3,3},{-3,3}}],{t,0,2 Pi}]

Manipulate[Show[ParametricPlot3D[{x,2 +Sin[x],0},{x,0,2Pi},PlotStyle->{{Thickness[0.01],RGBColor[0,0,1/2]}}],ParametricPlot3D[{x,(2 +Sin[x])Cos[t],(2 +Sin[x])Sin[t]},{x,0,2Pi},PlotStyle->{{Thickness[0.01],RGBColor[1/2,0,1/2]}}],
ParametricPlot3D[{x,(2 +Sin[x])Cos[s],(2 +Sin[x])Sin[s]},{x,0,2Pi},{s,0,t}],PlotRange->{{0,2Pi},{-3,3},{-3,3}},ImageSize->300],{t,0.001,2 Pi}]

Manipulate[Show[ParametricPlot3D[{θ-Sin[θ],1-Cos[θ],0},{θ,0,2Pi},PlotStyle->{{Thickness[0.01],RGBColor[0,0,1/2]}}],ParametricPlot3D[{θ-Sin[θ],(1-Cos[θ])Cos[t],(1-Cos[θ])Sin[t]},{θ,0,2Pi},PlotStyle->{{Thickness[0.01],RGBColor[1/2,0,1/2]}}],
ParametricPlot3D[{θ-Sin[θ],(1-Cos[θ])Cos[s],(1-Cos[θ])Sin[s]},{θ,0,2Pi},{s,0,t}],PlotRange->{{0,2Pi},{-3,3},{-3,3}},ImageSize->300],{t,0.001,2 Pi}]

ParametricPlot3D[(1-s){2Cos[θ]-Cos[2θ],2Sin[θ]-Sin[2θ],0}+s{0,0,3},{θ,0,2Pi},{s,0,2/4}]