(* This Mathematica file shows how to plot a cycloid and roll a circle *)

center[t_] := (({{t}, {0}}))
tanspacecoords[t_] := (({{Cos[-2t + Pi/2]}, {Sin[-2t + Pi/2]}}))
Flatten[center[t]+tanspacecoords[t]] (*This just changes how Mathematica understands the coordinates*)

ParametricPlot[Flatten[center[t]+tanspacecoords[t]],{t,0,8Pi},PlotRange->{{0,30},{-1,3}}]

cycloid:= ParametricPlot[Flatten[center[s]+tanspacecoords[s]],{s,0,8Pi},PlotRange->{{0,30},{-1,3}}]

circle[s_] := ParametricPlot[{Cos[t]+s, Sin[t]},{t,0,2Pi},PlotRange->{{0,30},{-1,3}}]
point[s_] := Graphics[Point[{Cos[-2s+Pi/2]+s, Sin[-2s + Pi/2]}]]

rollingcircle[s_] := Show[{cycloid, circle[s], point[s]}]

Manipulate[rollingcircle[s],{s,0,8Pi}]