Same Areas

p1[\[Theta]t_] :=
  With[{\[Theta] = Mod[\[Theta]t, 2 Pi]}, If[0 <= \[Theta] < Pi,
    {-Cos[\[Theta]], Sin[\[Theta]]},
    If[Pi <= \[Theta] < 3 Pi/2,
     0.5 {Cos[2 \[Theta]], Sin[2 \[Theta]]} + {0.5, 0}
     ,
     If[3 Pi/2 <= \[Theta] < 2 Pi,
      0.5 {-Cos[2 \[Theta]], -Sin[2 \[Theta]]} - {0.5, 0}
      ]
     ]]
   ];
n = 200;
p2[\[Theta]_] := 0.5 {-Cos[\[Theta]], Sin[\[Theta]]} - {0.5, 0};
p1s = Table[p1[\[Theta]], {\[Theta], 0, 2 Pi, 2 Pi/n}];
p2s = Table[p2[\[Theta]], {\[Theta], 0, 2 Pi, 2 Pi/n}];
ps0[t_] := t p1s + (1 - t) p2s;
ps[t_] := Sqrt[Pi] ps0[t]/Sqrt[Abs[area[ps0[t]]]];
area[pts_] :=
  Apply[Plus,
    Flatten[pts Map[({1, -1} Reverse[#] &), RotateLeft[pts]]]]/2;
\[Alpha] = -1;
Manipulate[
 With[{t = If[\[Tau] <= 1, \[Tau], \[Tau] - 1]},
  Graphics[
   {
    If[\[Tau] <= 1, White, {White, Disk[{0, 0}, 1.99], Black}],
    Rotate[Polygon[ps[t]], -\[Alpha] (1 - t), {-1, 0}],
    Rotate[
     Rotate[
      Polygon[ps[t]],
      -\[Alpha] (1 - t), {-1, 0}
      ]
     , Pi, {0, 0}]
    }
   , PlotRange -> 2, Background -> Black]
  ]
 , {\[Tau], 0, 2}, {invert, {True, False}}]