1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
|
## Copyright (C) 2024 David Legland
## All rights reserved.
##
## Redistribution and use in source and binary forms, with or without
## modification, are permitted provided that the following conditions are met:
##
## 1 Redistributions of source code must retain the above copyright notice,
## this list of conditions and the following disclaimer.
## 2 Redistributions in binary form must reproduce the above copyright
## notice, this list of conditions and the following disclaimer in the
## documentation and/or other materials provided with the distribution.
##
## THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS ''AS IS''
## AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
## IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
## ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE FOR
## ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
## DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
## SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
## CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
## OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
## OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
##
## The views and conclusions contained in the software and documentation are
## those of the authors and should not be interpreted as representing official
## policies, either expressed or implied, of the copyright holders.
function ell = equivalentEllipsoid(points)
%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
% ------
% Author: David Legland
% E-mail: david.legland@inrae.fr
% Created: 2011-03-12, using Matlab 7.9.0.529 (R2009b)
% Copyright 2011-2023 INRA - Cepia Software Platform
% number of points
n = size(points, 1);
% compute centroid
center = mean(points);
% compute the covariance matrix
covPts = cov(points)/n;
% perform a principal component analysis with 3 variables,
% to extract equivalent axes
[U, S] = svd(covPts);
% extract length of each semi axis
radii = sqrt(5) * sqrt(diag(S)*n)';
% sort axes from greater to lower
[radii, ind] = sort(radii, 'descend');
% format U to ensure first axis points to positive x direction
U = U(ind, :);
if U(1,1) < 0
U = -U;
% keep matrix determinant positive
U(:,3) = -U(:,3);
end
% convert axes rotation matrix to Euler angles
angles = rotation3dToEulerAngles(U);
% concatenate result to form an ellipsoid object
ell = [center, radii, angles];
|
