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## 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 = equivalentEllipse(points)
%EQUIVALENTELLIPSE Equivalent ellipse of a set of points.
%
% ELL = equivalentEllipse(PTS);
% Computes the ellips with the same moments up to the second order as the
% set of points specified by the N-by-2 array PTS.
%
% The result has the following form:
% ELL = [XC YC A B THETA],
% with XC and YC being the center of mass of the point set, A and B being
% the lengths of the equivalent ellipse (see below), and THETA being the
% angle of the first principal axis with the horizontal (counted in
% degrees between 0 and 180 in counter-clockwise direction).
% A and B are the standard deviations of the point coordinates when
% ellipse is aligned with the principal axes.
%
% Example
% pts = randn(100, 2);
% pts = transformPoint(pts, createScaling(5, 2));
% pts = transformPoint(pts, createRotation(pi/6));
% pts = transformPoint(pts, createTranslation(3, 4));
% ell = equivalentEllipse(pts);
% figure(1); clf; hold on;
% drawPoint(pts);
% drawEllipse(ell, 'linewidth', 2, 'color', 'r');
%
% See also
% ellipses2d, drawEllipse, equivalentEllipsoid, principalAxes,
% principalAxesTransform
%
% ------
% Author: David Legland
% E-mail: david.legland@inrae.fr
% Created: 2008-02-21, using Matlab 7.4.0.287 (R2007a)
% Copyright 2008-2023 INRA - BIA PV Nantes - MIAJ Jouy-en-Josas
% ellipse center
xc = mean(points(:,1));
yc = mean(points(:,2));
% recenter points
x = points(:,1) - xc;
y = points(:,2) - yc;
% number of points
n = size(points, 1);
% equivalent parameters
Ixx = sum(x.^2) / n;
Iyy = sum(y.^2) / n;
Ixy = sum(x.*y) / n;
% compute ellipse semi-axis lengths
common = sqrt( (Ixx - Iyy)^2 + 4 * Ixy^2);
ra = sqrt(2) * sqrt(Ixx + Iyy + common);
rb = sqrt(2) * sqrt(Ixx + Iyy - common);
% compute ellipse angle in degrees
theta = atan2(2 * Ixy, Ixx - Iyy) / 2;
theta = rad2deg(theta);
% create the resulting equivalent ellipse
ell = [xc yc ra rb theta];
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