Cycloids

Needs["Graphics`Colors`"];

circleplot[t_] := Graphics[Circle[{t,1},1]]
circleplot2[t_] := Graphics[Circle[{t,1},.5]]
edgepoint[t_] := Graphics[
                  {PointSize[.025],Red,
                   Point[{t-Sin[t],1-Cos[t]}]}]
edgepoint2[t_] := Graphics[
                  {PointSize[.025],Blue,
                   Point[{t/2-Sin[t],1/2-Cos[t]}]}]
origin[t_] := Graphics[
                {PointSize[.01],Point[{t,1}]}]
               
pointList = {};
pointList2 = {};

Table[
      AppendTo[pointList,edgepoint[t]];

      Show[circleplot[t],
           circleplot2[t],
           origin[t],
           pointList,

           PlotRange->{{-2,17},{-1,3}},
           Axes->True,
           AxesOrigin->{0,0},
           AspectRatio->Automatic],
   {t,0,4 Pi,Pi/6}];

Clear[r,x,y,t]
r = .5
x[theta_] = (t + (t-Sin[t]))/2
y[theta_] = (1 + (1-Cos[t]))/2

Needs["Graphics`Colors`"];

circleplot[t_] := Graphics[Circle[{t,1},1]]
circleplot2[t_] := Graphics[Circle[{t,1},.5]]
edgepoint[t_] := Graphics[
                  {PointSize[.025],Red,
                   Point[{t-Sin[t],1-Cos[t]}]}]
origin[t_] := Graphics[
                {PointSize[.01],Point[{t,1}]}]
edgepoint2[t_] := Graphics[
                  {PointSize[.025],Blue,
                   Point[{x[t],y[t]}]}]

pointList = {}
pointList2 = {}

Table[
      
      AppendTo[pointList2,edgepoint2[t]];
      AppendTo[pointList,edgepoint[t]];
	   Show[circleplot[t],
           circleplot2[t],
           origin[t],
           pointList,
           pointList2,
           PlotRange->{{-2,17},{-1,3}},
           Axes->True,
           AxesOrigin->{0,0},
           AspectRatio->Automatic,
           DisplayFunction->$DisplayFunction],
   {t,0,4 Pi,Pi/6}];