I have created the following code below. Its purpose is for me to divide the plot into grids and for each grid I reterive the center point’s co-ordinates.

`dp = DensityPlot[(E^-(x^2 + y^2)^2)^2 + ((E^-(x^2 + y^2)^2) (x^2 + y^2) Cos[2 Pi])^2, {x, -3, 3}, {y, -3, 3}, PlotTheme -> "Minimal", PlotRange -> All, PlotPoints -> 50, ColorFunction -> "Rainbow"]; ClearAll[centers2] centers2[region_ : Disk[{0, 0}, 1]][{nc_, nr_}, {xrange_, yrange_}] := Select[RegionMember[region]]@centers[{nc, nr}, {xrange, yrange}] {xrange, yrange} = {{-3, 3}, {-3, 3}}; {nc, nr} = {10, 10}; Show[dp, Graphics[{White, PointSize[Medium], Circle[], Point@centers2[][{nc, nr}, {xrange, yrange}]}]] means[n_] := MovingAverage[Subdivide[##, n] & @@ #, 2]; &; cp = centers2[][{nc, nr}, {xrange, yrange}] `

Next I have made a Do loop that will perform calcualtions to get a variable called angle. This code is given below.

`Do[point = Part[cp, i]; x = Part[point, 1]; y = Part[point, 2]; ex = (E^-(x^2 + y^2)^2)^2; ey = ((E^-(x^2 + y^2)^2) (x^2 + y^2) Cos[2 Pi])^2; s1 = ex^2 - ey^2; s2 = 2 ex ey Cos[0]; inside = 2 s2/s1; angle = ArcTan[inside], {i, Length[cp]}] `

Now what I would like to have a command inside the Do loop that I can use to plot an ellipse based on the variable angle (this will be used to give the angle that the semi major axis makes with the x axis) for very iteration of the Do loop. There is a constant semi and minor axis. The point the ellipse will have to be centered at is the point that is currently being used in the Do loop.

I am not sure how to do this inside the loop for all the iterations, however I have a found a method to plot an ellipse given as

`ellipsoid[center_, {majorradius_, minorradius_}, angle_] := GeometricTransformation[ Ellipsoid[center, {majorradius, minorradius}], RotationTransform[angle, center]] ad = Pi/3; sd = Graphics[{EdgeForm[{Thick}], Opacity[.75], Transparent, ellipsoid[{0, 0}, {0.2, 0.1}, ad], Opacity[1], Red, Point[{0, 0}],}] Plot[3, sd, {x, -10, -10}] `

I also would like to store these plots as I would like to then combine all of these ellipse plots onto the densityplot I have made in the beginning

All help is much appreciated