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graphics - Plot Ellipse based on EigenSystem



Please consider the following :


covMatrice = {{34.925, -10.21}, {-10.21, 22.462}};

COG = Mean@(Nfixations["d"][[1]])[[All, 1]];

fixations = {{19.4688, 17.4281}, {18.0563, 21.7156}, {13.0219, 24.7219}, 
         {22.9594,25.5219}, {28.5406, 24.6719}, {27.0688, 17.1656}, 
         {27.6781,16.325}, {28.9281, 10.7719}, {16.025, 13.6625}, 
         {19.1313, 17.1094}};



With[{
      eigVec = Eigenvectors[covMatrice],
      eigVal = Eigenvalues[covMatrice]},
      Graphics[{
                White, Rectangle @@ frmXY, 
                Black, Disk[#, .5] & /@ fixations, 
                Red, Line[(# + COG) & /@
                           {eigVec[[1]]*eigVal[[1]]/5, 
                           {0, 0},

                           eigVec[[2]]*eigVal[[2]]/5}]}]]

enter image description here


How could I draw an ellipse representing on the EigenSystem given that Neither Disk or Circle enable to implement an "orientation" ?


A rough example of my desired output as drawn using PPT :


enter image description here



Answer



You can use Rotate to draw the ellipse too. Note that knowing the eigenvectors is the same as knowing the orientation of the ellipse, so there is no necessity to favour GeometricTransformation over Rotate. So, to orient your ellipse along the first eigenvector (corresponding to the largest eigenvalue), it is as simple as:


Graphics[Rotate[Disk[meanVec, eigVals], ArcTan @@ eigVecs[[1]]]]


enter image description here


where meanVec is the mean (here, I've taken it to be {0,0})



Going by your comment under rcollyer's answer, here's a example reproducing your desired figure with simple shifting (change the center of the disk) and rotation of the disk. This approach will be simpler to follow (as the transformations are spelled out), if you do not understand what GeometricTransformation does. Modifying rcollyer's module,


Module[{mat = #, avg = Mean@fixations, eigVals, eigVecs},
   {eigVals, eigVecs} = Eigensystem@mat;
   Graphics[{{Black, Disk[#, .5] & /@ fixations}, 
      {Directive[Opacity[0.1], Red, EdgeForm[Gray]], 
          Rotate[Disk[avg, eigVals/5], ArcTan @@ eigVecs[[1]]]}
   }]

]&@covMatrice

enter image description here


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