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I guess there is some floating-point-related issue here... This works: f[t_] := RegionPlot[ TransformedRegion[ Rectangle[{-1, -1}, {1, 1}], { Indexed[#1, {1}] (1 + t (Indexed[#1, {2}]^2 - 1)) + 2 t, Indexed[#1, {2}] (1 + t (Indexed[#1, {1}]^2 - 1)) + 2 t } & ], PlotRange -> {{-1 + 2 t, 1 + 2 t}, {-1 + 2 t, 1 + 2 t}} ] ...



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