MATLAB: How to maintain aspect ratio of filled ellipses when using "fill" function?

Question

I am plotting filled circles and ellipses on an x-y plot. The ellipses are derived from 2x2 tensor data. The x-direction and y-direction have wildly different units depending on the type of data I am plotting. I want an ellipse to be plotted at some (x,y) location on the plot, but I want the aspect ratio of the plotted ellipse to be maintained regardless of the x and y axis units. Note that axis equal is not an option here because the x and y scales are so different. If I try to do axis equal it just makes the plot really tiny.

Below is a simple example. In this example, the first ellipse is always a perfect circle for reference to see how the figure distorts it. Any help is appreciated.

x = 100*[1 2 3 4]; %x direction is several orders of magnitude larger than y direction
y = [1 2 3 4]; %y direction
data = randn(4,1); %data to fill ellipse (irrelevant to the question)

nrot = 36; %Number of points on each ellipse

for i = 1:4
    %Elements of 2x2 tensor to make the ellipse
    r1 = randn; r2 = randn; r3 = randn; r4 = randn;

    for irot=1:nrot %Loop to get ellipse points

        rot_ang = (irot-1)*360/(nrot-1);
        v = [cosd(rot_ang),sind(rot_ang)]; %Rotating vector components

        if i == 1 %Ensure that the first ellipse is a perfect circle
            r1 = 0; r4 = 0; r3 = r2;
        end

        plot_vec = [r1 r2; r3 r4]*v';

        %x component of ellipse to plot
        ex(irot) = plot_vec(1);

        %y component of ellipse to plot
        ey(irot) = plot_vec(2);  
    end 

    %Plot the ellipse at the x-y location
    xp = x(i)+ex;
    yp = y(i)+ey;

    fill(xp,yp,data(i)); hold on %Plot ellipses filled with "data".


end

%"Axis equal" does not work in this case
%axis equal

Answer 1

It sounds like you want your ellipses to have a displayed data aspect ratio of 1:1, even when the data aspect ratio of the axes is not. One option is to first choose the data aspect ratio you want for your axes, then scale the y values of your ellipses accordingly before translating and plotting them:

x = 100*[1 2 3 4];
y = [1 2 3 4];
aspectRatio = 100;  % Choose an aspect ratio of 100
data = randn(4, 1);
nrot = 36;

for i = 1:4

  r1 = randn; r2 = randn; r3 = randn; r4 = randn;

  for irot = 1:nrot

    rot_ang = (irot-1)*360/(nrot-1);
    v = [cosd(rot_ang), sind(rot_ang)];

    if i == 1
      r1 = 0; r4 = 0; r3 = r2;
    end

    plot_vec = [r1 r2; r3 r4]*v';
    ex(irot) = plot_vec(1);
    ey(irot) = plot_vec(2);

  end

  xp = x(i)+ex;
  yp = y(i)+ey./aspectRatio;  % Scale ellipse y data by aspect ratio

  fill(xp, yp, data(i));
  hold on;
  daspect(gca, [aspectRatio 1 1]);  % Set aspect ratio of axes

end

And here's the resulting plot:

The ellipses are small, but if you zoom you'll see they appear to have the correct aspect ratio (i.e. the first looks like a circle).