The wandering Ellipse

Now I will describe my code. Create a plot of the function, which is given parametrically

f[t_] := {t,Sin[t]} 
ParametricPlot[f[t],{t,-2Pi,2},PlotRange->{{-8,4},{-2,2}},PlotStyle->Red]

Create a new function

g[t_] := (f[t0]+f''[t0])+(f'[t0] Cos[t0]-f''[t0] Sin[t0]) Sin[t]-(f'[t0] Sin[t0]+f''[t0] Cos[t0]) Cos[t]

Create animation

Animate[Show[ParametricPlot[f[t],{t,-2.3 Pi,E},PlotRange->{{-8,Pi},{-2,2}},
PlotStyle->{Thick,Red},Axes->False],
ParametricPlot[f[t], {t,k-0.7,k+0.65},
PlotStyle->Directive[Green, Thickness[0.014]]],
ParametricPlot[g[t] /.t0 -> k,{t,-5,2Pi},PlotStyle->Black]],{k,-6.8,2,.1}]

You can still add the second ellipse

Created by Silva Torosyan in Wolfram Language