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computational geometry - Can't plot rotated region


I was experimenting with the code from this question when I ran into another problem with regions.


Ω = RegionDifference[Rectangle[{0, 0}, {10, 10}], Rectangle[{4, 4}, {8, 8}]];
RegionPlot[Ω]

plot


Ω1 = TransformedRegion[Ω, RotationTransform[45 °, {5, 5}]];
RegionQ[Ω1]



True

RegionPlot[Ω1]


RegionPlot::invplotreg: TransformedRegion[RegionDifference[Rectangle[{0, 0}, {10, 10}], Rectangle[{4, 4}, {8, 8}]], TransformationFunction[...]] is not a valid region to plot. >>



What is the difference between a "valid region to plot" and a region that satisfies RegionQ? Or, perhaps, to put it better, am I seeing a bug in RegionPlot or just an incomplete implementation?



I note that


RegionPlot[TransformedRegion[Rectangle[], RotationTransform[45 °, {.5, .5}]], 
PlotRange -> All]

works as expected, so it would seem RegionPlot can handle rotations for some class of inputs.



Answer



Here is a workaround:


r1 = RegionDifference[Rectangle[{0, 0}, {10, 10}], 
Rectangle[{4, 4}, {8, 8}]];
r2 = TransformedRegion[r1, RotationTransform[45 \[Degree], {5, 5}]];

mr = DiscretizeRegion[r2]

enter image description here


And then:


RegionPlot[mr]

enter image description here


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