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## Copyright (C) 2004-2011 David Legland <david.legland@grignon.inra.fr>
## Copyright (C) 2004-2011 INRA - CEPIA Nantes - MIAJ (Jouy-en-Josas)
## Copyright (C) 2012 Adapted to Octave by Juan Pablo Carbajal <carbajal@ifi.uzh.ch>
## 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.
## -*- texinfo -*-
## @deftypefn {Function File} {@var{ell} = } inertiaEllipse (@var{pts})
## Inertia ellipse of a set of points
##
## ELL = inertiaEllipse(PTS);
## where PTS is a N*2 array containing coordinates of N points, computes
## the inertia ellispe of the set of points.
##
## The result has the form:
## ELL = [XC YC A B THETA],
## with XC and YC being the center of mass of the point set, A and B are
## the lengths of the inertia ellipse (see below), and THETA is the angle
## of the main inertia axis with the horizontal (counted in degrees
## between 0 and 180).
## A and B are the standard deviations of the point coordinates when
## ellipse is aligned with the inertia 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 = inertiaEllipse(pts);
## figure(1); clf; hold on;
## drawPoint(pts);
## drawEllipse(ell, 'linewidth', 2, 'color', 'r');
## @end example
##
## @seealso{ellipses2d, drawEllipse}
## @end deftypefn
function ell = inertiaEllipse(points)
# 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);
# inertia parameters
Ixx = sumsq (x) / n;
Iyy = sumsq (y) / 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 inertia ellipse
ell = [xc yc ra rb theta];
endfunction
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