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The issue at least on my machine (Linux, M10.1) is that the plots take a long time to generate and so creating a smooth animation with on-the-fly generated plots is, well, impossible. You could generate the plots beforehand, though. Here's some code to show you the progress of plot generation as well. Assuming that you have defined a function f[t] that ...


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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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