In[]:=
negateFirstRow[A_]:={-A[[1,All]]}~Join~A[[2;;,All]];​​sampleO[n_]:=RandomVariate@CircularRealMatrixDistribution@n;​​sampleSO[n_]:=With[{mat=sampleO@n},If[Det[mat]>0,mat,negateFirstRow@mat]];​​​​n=4;​​numSamples=20000;​​data=Flatten@Table[Eigenvalues@sampleSO@n,{numSamples}];​​ComplexListPlot[data,PlotStyle->{PointSize[Large],Opacity[.001]}]
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-1.0
-0.5
0.5
1.0
-1.0
-0.5
0.5
1.0
​
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(*createsmoothkerneldistributionofArgs*)​​args=Arg@data;​​skd=SmoothKernelDistribution[args]​​Plot[PDF[skd,θ],{θ,-Pi,Pi}]​​​​
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DataDistribution
Type: SmoothKernel
Data points: 80000

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(*create3dsectorchartbasedonPDFofthesmooothkerneldistribution*)​​dθ=Pi/50;​​createSector[θ_]:={dθ,1,PDF[skd,θ]}​​​​rangeArguments=Join[MinMax[args],{dθ}];​​thetaRange=Range@@rangeArguments;​​sectorList=createSector/@thetaRange;
In[]:=
SectorChart3D[sectorList]
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