\(\(\(
\)\(\(FillColor\  = \ 
        RGBColor[1,  .75,  .75];\)

    << Graphics`InequalityGraphics`

    \($TextStyle = {FontFamily -> "Times", FontSize -> 10, 
          FontWeight -> "Bold", 
          FontSlant -> "Italic"};\)

 (*\ Define\ B\ \ and\ the\ polar\ function\ XBX\ *) 

    B\  := \ {{lambda1, \ \((lambda2\  - \ lambda1)\) Cot[delta]}, {0, 
          lambda2}}

    XBX[theta_]\  := \ {Cos[theta], Sin[theta]} . 
        B . {Cos[theta], 
          Sin[theta]}

 (*\ 
      The\ eigenvalues\ to\ be\ used\ for\ this\ example\ *) 

    \(r = \ {lambda1 -> \(-1\), 
          lambda2 -> \(-5\)};\)


    \(animation\  = \ 
        Table[
Graphics[
            Show[

 (*\ 
                Shade\ the\ regions\ where\ x . 
                    Bx\ is\ positive\ *) 
InequalityPlot[{x1, x2} . B . {x1, x2}\  > \ 0\  /. \ 
                  r, \ {x1, \(- .5\),  .5}, {x2, \(- .5\),  .5}, \ 
                Ticks -> \ None, 
                PlotRange -> {{\(- .6\),  .6}, {\(- .6\),  .6}}, 
                AspectRatio -> 1, \ 
                DisplayFunction\  -> \ Identity, \ 
                AxesLabel -> {\*""(Subscript[x, 1])"", \*""(Subscript[x, 2])""}, 
                Fills\  -> \ 
                  FillColor], 

 (*\ 
                Scaled\ polar\ plot\ of\ x . 
                    Bx\ *) 
ParametricPlot[\((XBX[theta]/\ 
                        Max[Abs[\((lambda1 + 
                                lambda2 + \((lambda2 - lambda1)\)\ Csc[
                                    delta])\)], 
                          Abs[\((lambda1 + 
                                lambda2 - \((lambda2 - lambda1)\)\ Csc[
                                    delta])\)]])\) {Cos[theta], 
                      Sin[theta]}\  /. \ r, {theta, 0, 2  Pi}, 
                PlotStyle -> \ {Thickness[ .0075]}, 
                DisplayFunction\  -> \ 
                  Identity], 

 (*\ 
                Plot\ the\ eigenvector\ corresponding\ to\ lambda2\ *) 
ParametricPlot[
                t {Cos[delta], Sin[delta]}\ , {t, 0,  .4}, 
                PlotStyle -> {RGBColor[0, 0, 1], Thickness[ .01]}, 
                DisplayFunction\  -> \ 
                  Identity], 

 (*\ 
                Plot\ evector\ corresponding\ to\ lambda1, \ 
                always\ horizontal\ *) 
ParametricPlot[{t \((lambda1/lambda2)\), 0}\  /. r, {t, 0,  .4}, 
                PlotStyle -> {RGBColor[0, 0, 1], Thickness[ .01]}, 
                DisplayFunction\  -> \ 
                  Identity]
]], 

 (*\ 
            We\ need\ to\ skip\ 0\ and\ 2  
              Pi\ with\ delta\ since\ there\ are\ discontinuities\ there\ and\ B\ does\ not\ exist\ *) 
{delta,  .0125  Pi, 
            2  Pi -  .0125  Pi,  .025  
              Pi}];\)

 (*\ 
      Export\ the\ animation\ as\ a\ gif\ *) 

    Export["bubbles_real.gif", animation, 
      ConversionOptions -> {"Loop" -> 
            True, "AnimationDisplayTime" -> 0.1}, 
      ImageSize -> {250, 250}, 
      ImageResolution -> 144]

 (*\ 
      Export\ a\ snapshot\ to\ use\ as\ a\ thumbnail\ *) 

    Export["bubbles_real_tn.gif", animation[\([6]\)]]

    \)\)\)