Plotting the solutions of an equation (rotated ellipsoid)

Here is the equaion

one equation with three unknowns, so we need to give two variables to get the third one. Here is what I have done to solve the equation and plot the solutions:

Eqn = 4*x^2 - (y - 1/3)*((3*3^(1/2)*(z - 1/6))/16 - (13*y)/16 + 13/48) - (z - 1/6)*((3*3^(1/2)*(y - 1/3))/16 - (7*z)/16 + 7/96) == 1;

S = solve(Eqn, [z, x, y], 'ReturnConditions', true);

Av1 = -.6; Bv1 = .6; Nv1=15; Av2 = -1; Bv2 = 2; Nv2=20;

[iv1, iv2] = meshgrid(linspace(Av1, Bv1, Nv1), linspace(Av2, Bv2, Nv2));

iv1 = iv1(:);

iv2 = iv2(:);

v1 = zeros(2,1); v2 = zeros(2,1); v3 = zeros(2,1);

sv1 = zeros(2,1); sv2 = zeros(2,1); sv3 = zeros(2,1);

counter = 1;

for i = 1: length(iv1)

Zv = subs(S.zv, S.parameters, [iv2(i), iv1(i)]);

if isreal(Zv(1))

v1(counter) = iv1(i);

v2(counter) = iv2(i);

v3(counter) = max(Zv);

sv1(counter) = iv1(i);

sv2(counter) = iv2(i);

sv3(counter) = min(Zv);

counter = counter +1;

end

end

V = [v1, v2, v3]; %these are the points on the upper part of rotated ellipsiod

sV = [sv1, sv2, sv3]; %these are the points on the upper part of rotated ellipsiod

v1lin = linspace( min(v1), max(v1), 100);

v2lin = linspace(min(v2), max(v2), 100);

[Xv, Yv] = meshgrid(v1lin, v2lin);

Zv = griddata(v1,v2,v3,Xv,Yv, 'cubic');

sv1lin = linspace( min(sv1), max(sv1), 100);

sv2lin = linspace(min(sv2), max(sv2), 100);

[sXv, sYv] = meshgrid(sv1lin, sv2lin);

sZv = griddata(sv1,sv2,sv3,sXv,sYv, 'cubic');

figure

mesh(Xv, Yv, Zv)

xlabel('Xv')

ylabel('Yv')

zlabel('Zv')

hold on

axis tight

mesh(sXv, sYv, sZv)

title('The complete surface by using Griddata interpolate on V vectors')

figure

mesh(Xv, Yv, Zv)

xlabel('Xv')

ylabel('Yv')

zlabel('Zv')

hold on

axis tight

title('The half surface by using Griddata interpolate on V vectors')

As the obtained solutions are not uniformly distributed, I have used the griddata() to generate some uniformly spaced points to be able to plot the desired surface. However, the problem is that I cannot get to the complete rotated ellipsoid. Please see the attached photos. The blue points are the points which have been obtained by solving the equation. From mathematics, I know that the equation above is a rotated ellipsoid, but I cannot get there and always there is some part missing.

Is there any idea on how I can find some more points on the ellipsoid by solving the equation (or using some other techniques) which help me to plot the entire ellipsoid?

I should say that I have asked this question in another way here. However, as it has not received any notable attention, I tried to ask my question in a new way here.

Any help would be appreciated.

Answer 1

[Matt J]

If you have the center, radii, and yaw-pitch-roll angles (in degrees) of the ellipsoid, you can plot a data-free ellipsoidalFit object using,

https://www.mathworks.com/matlabcentral/fileexchange/87584-object-oriented-tools-for-fitting-conics-and-quadrics

The final section in the Examples tab at this link gives examples of this, but basically it is,

gtEllip=ellipsoidalFit.groundtruth([],center,radii,yaw_pitch_roll);

plot(gtEllip)

[Majid]

Matt, thank you so much for sharing your interesting tool set. I do not know, so far, how I can calculate the yaw-pitch-roll angles from the equation of an ellipsoid. Solving that, I would like to know if it is possible to set the color in your tool set, or if it is possible to have the figure as a mesh? or to add the axis at the origin?

[Matt J]

it is possible to set the color in your tool set

Yes, definitely. Plenty of examples of that at the link I gave you.

or if it is possible to have the figure as a mesh

Not directly, but you can use the gtEllip.sample() method to get surface points on the ellipse and then use alphaShape() to get a mesh, as we talked about in another thread today.

or to add the axis at the origin?

You can use line() to add lines to the plot.

[Majid]

Thank you Matt for your explanations! As always, very helpful! I could not find out the yaw-pitch-roll angles. Actually, my main aim is not to plot the solutions of the above equation, but it is to plot the solutions of a system of differential equations in which the solutions of the above equation are used as initial conditions. However, as for a special case, the solutions of the system and the equation were the same, so I asked the question here in this way.

Anyway, your reply to my comment in another thread worked very well and saved me! Thank you so much!!!

If it is possible, I would like to know what your field of research is.