Affine Transformation of a Region 

Apply affine transformations to basic geometric regions.

In[1]:=

X t = ShearingTransform[30 Degree, {1, 0}, {0, 1}];

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X sheared = TransformedRegion[Rectangle[], t];

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X RegionPlot[sheared, PlotRange -> All]

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Affine transformations can also be applied to formula regions.

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X \[ScriptCapitalR] = ImplicitRegion[ x^2 + y^2 + z^2 < 1 || (x + 1)^2 + y^2 + z^2 < 1, {x, y, z}];

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X scaled = TransformedRegion[\[ScriptCapitalR], ScalingTransform[{1, 2, 2}]];

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X DiscretizeRegion /@ {\[ScriptCapitalR], scaled}

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