/usr/share/octave/packages/geometry-1.7.0/geom2d/inertiaEllipse.m is in octave-geometry 1.7.0-1build1.
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## 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
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## -*- 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 = 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 inertia ellipse
ell = [xc yc ra rb theta];
endfunction
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