equivalentEllipsoid

 Equivalent ellipsoid of a set of 3D points.

   ELL = equivalentEllipsoid(PTS)
   Compute the equivalent ellipsoid of the set of points PTS. The result
   is an ellipsoid defined by:
   ELL = [XC YC ZC  A B C  PHI THETA PSI]
   where [XC YC ZY] is the center, [A B C] are the lengths of the
   semi-axes (in decreasing order), and [PHI THETA PSI] are Euler angles
   representing  the ellipsoid orientation, in degrees.

   Example
     pts = randn(300, 3);
     pts = transformPoint3d(pts, createScaling3d([6 4 2]));
     pts = transformPoint3d(pts, createRotationOx(pi/6));
     pts = transformPoint3d(pts, createRotationOy(pi/4));
     pts = transformPoint3d(pts, createRotationOz(pi/3));
     pts = transformPoint3d(pts, createTranslation3d([5 4 3]));
     elli = equivalentEllipsoid(pts);
     figure; drawPoint3d(pts); axis equal;
     hold on; drawEllipsoid(elli, ...
         'drawEllipses', true, 'EllipseColor', 'b', 'EllipseWidth', 3);

   See also
     spheres, drawEllipsoid, equivalentEllipse, principalAxes
     principalAxesTransform, rotation3dToEulerAngles

0001 function ell = equivalentEllipsoid(points)
0002 % Equivalent ellipsoid of a set of 3D points.
0003 %
0004 %   ELL = equivalentEllipsoid(PTS)
0005 %   Compute the equivalent ellipsoid of the set of points PTS. The result
0006 %   is an ellipsoid defined by:
0007 %   ELL = [XC YC ZC  A B C  PHI THETA PSI]
0008 %   where [XC YC ZY] is the center, [A B C] are the lengths of the
0009 %   semi-axes (in decreasing order), and [PHI THETA PSI] are Euler angles
0010 %   representing  the ellipsoid orientation, in degrees.
0011 %
0012 %   Example
0013 %     pts = randn(300, 3);
0014 %     pts = transformPoint3d(pts, createScaling3d([6 4 2]));
0015 %     pts = transformPoint3d(pts, createRotationOx(pi/6));
0016 %     pts = transformPoint3d(pts, createRotationOy(pi/4));
0017 %     pts = transformPoint3d(pts, createRotationOz(pi/3));
0018 %     pts = transformPoint3d(pts, createTranslation3d([5 4 3]));
0019 %     elli = equivalentEllipsoid(pts);
0020 %     figure; drawPoint3d(pts); axis equal;
0021 %     hold on; drawEllipsoid(elli, ...
0022 %         'drawEllipses', true, 'EllipseColor', 'b', 'EllipseWidth', 3);
0023 %
0024 %   See also
0025 %     spheres, drawEllipsoid, equivalentEllipse, principalAxes
0026 %     principalAxesTransform, rotation3dToEulerAngles
0027 
0028 % ------
0029 % Author: David Legland
0030 % e-mail: david.legland@inrae.fr
0031 % Created: 2011-03-12,    using Matlab 7.9.0.529 (R2009b)
0032 % Copyright 2011 INRA - Cepia Software Platform.
0033 
0034 % number of points
0035 n = size(points, 1);
0036 
0037 % compute centroid
0038 center = mean(points);
0039 
0040 % compute the covariance matrix
0041 covPts = cov(points)/n;
0042 
0043 % perform a principal component analysis with 3 variables,
0044 % to extract equivalent axes
0045 [U, S] = svd(covPts);
0046 
0047 % extract length of each semi axis
0048 radii = sqrt(5) * sqrt(diag(S)*n)';
0049 
0050 % sort axes from greater to lower
0051 [radii, ind] = sort(radii, 'descend');
0052 
0053 % format U to ensure first axis points to positive x direction
0054 U = U(ind, :);
0055 if U(1,1) < 0
0056     U = -U;
0057     % keep matrix determinant positive
0058     U(:,3) = -U(:,3);
0059 end
0060 
0061 % convert axes rotation matrix to Euler angles
0062 angles = rotation3dToEulerAngles(U);
0063 
0064 % concatenate result to form an ellipsoid object
0065 ell = [center, radii, angles];