Deriving rotation angle from the eigenvectors

For an ellipsoid, how can I get the rotation angle from the eigenvectors?

[James Tursa]

Please give more details. The rotation angle from what to what? Are you looking for a single angle of some type, or a 3D rotation, or ...?

[Sophie Wuerger]

I think I just found the solution for the 3d rotation.

[center,radii,ecvecs,v,chi2]=ellipsoid_fit(dat,'');
[x, y, z] = ellipsoid(center(1), center(2),center(3), radii(1),radii(2),radii(3),30);
S=surf(x, y, z,'FaceAlpha', 0.3, 'EdgeColor', 'none', 'FaceLighting', 'gouraud')
.....
    theta1 = -asind(R(3,1)); % y-axis 
psi1 = atan2d(R(3,2)/cos(theta1),R(3,3)/cos(theta1)); % x-axis 
phi1 = atan2d(R(2,1)/cos(theta1),R(1,1)/cos(theta1)); % z-axis 
direction = [1 0 0]; %
rotate(S,direction,psi1,center)
direction = [0 1 0]; 
rotate(S,direction,theta1,center)
direction = [0 0 1]; 
rotate(S,direction,phi1,center)

Does that look allright?

Is there an easy way to obtain the projection onto the axes?

best

sophie

[Qingda Hu]

Probably too late but I was trying to use the same ellipsoid_fit function

If you linearize x,y,z and just multiple it by the evecs you get what you want

[center, radii, evecs, pars ] = ellipsoid_fit_new(... );

[X,Y,Z] = ellipsoid(0,0,0,radii(1),radii(2),radii(3));

Xlin = X(:);

Ylin = Y(:);

Zlin = Z(:);

rotatedXYZ = [Xlin,Ylin,Zlin]*evecs';

X = reshape( rotatedXYZ(:,1),size(X));

Y = reshape( rotatedXYZ(:,2),size(X));

Z = reshape( rotatedXYZ(:,3),size(X));

surf(X,Y,Z);

(you can translate it as well)

Hopefully this is useful for someone.

Answer 1

[Matt J]

If you download ellipsoidalFit() to do the fitting, the yaw-pitch-roll angles are avaialble directly from the object,

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

>> load data
>> fitobj=ellipsoidalFit(xyz)

fitobj = 

  ellipsoidalFit with properties:

    center: [5.0483 2.0378 7.0195]
         a: 20.0700
         b: 10.0469
         c: 4.9622
       yaw: 39.6951
     pitch: -29.7995
      roll: 50.4333