1## Copyright (C) 2021 David Legland
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27
28function ell = inertiaEllipse(points)
29%INERTIAELLIPSE Inertia ellipse of a set of points.
30%
31%   Note: Deprecated! Use equivalentEllipse instead.
32%
33%   ELL = inertiaEllipse(PTS);
34%   where PTS is a N*2 array containing coordinates of N points, computes
35%   the inertia ellipse of the set of points.
36%
37%   The result has the form:
38%   ELL = [XC YC A B THETA],
39%   with XC and YC being the center of mass of the point set, A and B are
40%   the lengths of the inertia ellipse (see below), and THETA is the angle
41%   of the main inertia axis with the horizontal (counted in degrees
42%   between 0 and 180).
43%   A and B are the standard deviations of the point coordinates when
44%   ellipse is aligned with the principal axes.
45%
46%   Example
47%   pts = randn(100, 2);
48%   pts = transformPoint(pts, createScaling(5, 2));
49%   pts = transformPoint(pts, createRotation(pi/6));
50%   pts = transformPoint(pts, createTranslation(3, 4));
51%   ell = inertiaEllipse(pts);
52%   figure(1); clf; hold on;
53%   drawPoint(pts);
54%   drawEllipse(ell, 'linewidth', 2, 'color', 'r');
55%
56%   See also
57%     equivalentEllipse
58%
59
60% ------
61% Author: David Legland
62% e-mail: david.legland@inra.fr
63% Created: 2008-02-21,    using Matlab 7.4.0.287 (R2007a)
64% Copyright 2008 INRA - BIA PV Nantes - MIAJ Jouy-en-Josas.
65
66% HISTORY
67% 2009-07-29 take into account ellipse orientation
68% 2011-03-12 rewrite using inertia moments
69
70% deprecation warning
71warning('geom2d:deprecated', ...
72    [mfilename ' is deprecated, use ''equivalentEllipse'' instead']);
73
74% ellipse center
75xc = mean(points(:,1));
76yc = mean(points(:,2));
77
78% recenter points
79x = points(:,1) - xc;
80y = points(:,2) - yc;
81
82% number of points
83n = size(points, 1);
84
85% inertia parameters
86Ixx = sum(x.^2) / n;
87Iyy = sum(y.^2) / n;
88Ixy = sum(x.*y) / n;
89
90% compute ellipse semi-axis lengths
91common = sqrt( (Ixx - Iyy)^2 + 4 * Ixy^2);
92ra = sqrt(2) * sqrt(Ixx + Iyy + common);
93rb = sqrt(2) * sqrt(Ixx + Iyy - common);
94
95% compute ellipse angle in degrees
96theta = atan2(2 * Ixy, Ixx - Iyy) / 2;
97theta = rad2deg(theta);
98
99% create the resulting inertia ellipse
100ell = [xc yc ra rb theta];
101