%% Copyright (c) 2011, INRA
%% 2008-2011, David Legland <david.legland@grignon.inra.fr>
%% 2011 Adapted to Octave by Juan Pablo Carbajal <carbajal@ifi.uzh.ch>
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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

