Manipulate[ptsx=Table[{t,0},{t,0,1,1/(n-1)}];
ptsy=Table[{0,f[t]},{t,0,1,1/(n-1)}];
Graphics[{Black,PointSize[0.015],Point/@Join[ptsx,ptsy],Red,Table[Line[{ptsx[[i]],ptsy[[i]]}],{i,Length[ptsx]}]},PlotRange->{{-0.05,1.05},{-0.05,1.05}},ImageSize->500],{{n,40,"Número de Clavos"},5,150,1},{{f,Function[t,1-t],"Función"},{Function[t,1-t]->"1-t",Function[t,1-t^2]->"1-t^2",Function[t,Sqrt[1-t]]->"sqrt(1-t)",Function[t,Exp[-3 t]]->"exp(-3t)",Function[t,Sin[Pi t/2]]->"sin(t)",Function[t,Cos[Pi t/2]]->"cos(t)"}}]

h[c_,t_]:=c E^(I t)+E^(c I t)//ReIm
curve=ParametricPlot[h[2,t],{t,0,2Pi}]

Manipulate[pts=Table[{Cos[2 Pi k/n],Sin[2 Pi k/n]},{k,0,n-1}];
Graphics[{Black,PointSize[0.015],Point/@pts,Red,Table[Line[{pts[[k+1]],pts[[Mod[rule[k],n]+1]]}],{k,0,n-1}]},ImageSize->500],{{n,40},5,250,1},{{rule,Function[k,2 k],"Regla"},{Function[k,2 k]->"2k",Function[k,4k]->"4k",Function[k,100k]->"100k",Function[k,8k]->"8k"}}]

cardioide=ReIm/@Table[2E^(I s)+E^(2I s),{s,0,2 Pi,.01}];

Manipulate[Graphics[{Circle[{0,0},1],Circle[ReIm[2Exp[I t]]],Red,Line[cardioide],Black,PointSize[Large],Point[ReIm[2Exp[I t]+Exp[2I t]]]},PlotRange->5,Axes->True,ImageSize->600],{t,0,2 Pi}]

Manipulate[pts=Table[{Cos[2 Pi k/n],Sin[2 Pi k/n]},{k,0,n-1}];
Graphics[{PointSize[0.006],Black,Point/@pts,Blue,Table[Line[{pts[[k+1]],pts[[Mod[m k,n]+1]]}],{k,0,n-1}]},ImageSize->500],{{n,50},0,500,10},{{m,2},2,20,1}]

Manipulate[pts=Table[{Cos[2 Pi k/n],Sin[2 Pi k/n]},{k,0,n-1}];
Graphics[{Black,PointSize[0.015],Point/@pts,Red,Table[Line[{pts[[k+1]],pts[[Mod[rule[k],n]+1]]}],{k,0,n-1}]},ImageSize->500],{{n,40},5,250,1},{{rule,Function[k,2 k],"Regla"},{Function[k,2 k]->"2k",Function[k,3k]->"3k",Function[k,10k]->"10k",Function[k,20k]->"20k"}}]