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}}]

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