(* Parametric plot of circle *)
P[t_]:={Cos[t],Sin[t]};

(* Barbell contour *)
B[t_]:= If[t<Pi/3,P[3t]+{2,0}, If[Pi/3≤ t < 2Pi/3,{2(Pi/3-t)3/Pi+1,0},
If[ 2Pi/3≤t<4Pi/3,P[3t]-{2,0},
If[4Pi/3≤ t<5Pi/3,{2(t-4Pi/3)3/Pi-1,0},
If[5Pi/3≤t,P[(t-4Pi/3)3]+{2,0}]]]
]]

(* The homotopy *)

H[s_,t_]:= s B[t] +(1-s)4 P[t]

(* Some intermediate curves *)

ParametricPlot[{H[0,t],H[.3,t],H[.6,t],H[1,t]},{t,0,2Pi}]

(* Animation *)
Animate[ParametricPlot[{H[0,t],H[s,t],H[1,t]},{t,0,2 π},PlotStyle->{Blue,Red,Black}] ,{s,0,1},AnimationRunning->False]