% PLOTCOV2 - Plots a covariance ellipse with major and minor axes
%            for a bivariate Gaussian distribution.
%
% Usage:
%   h = plotcov2(mu, Sigma[, OPTIONS]);
% 
% Inputs:
%   mu    - a 2 x 1 vector giving the mean of the distribution.
%   Sigma - a 2 x 2 symmetric positive semi-definite matrix giving
%           the covariance of the distribution (or the zero matrix).
%
% Options:
%   'conf'    - a scalar between 0 and 1 giving the confidence
%               interval (i.e., the fraction of probability mass to
%               be enclosed by the ellipse); default is 0.9.
%   'num-pts' - the number of points to be used to plot the
%               ellipse; default is 100.
%
% This function also accepts options for PLOT.
%
% Outputs:
%   h     - a vector of figure handles to the ellipse boundary and
%           its major and minor axes
%
% See also: PLOTCOV3

% Copyright (C) 2002 Mark A. Paskin
%
% This program is free software; you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation; either version 2 of the License, or
% (at your option) any later version.
%
% This program is distributed in the hope that it will be useful, but
% WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
% General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this program; if not, write to the Free Software
% Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307
% USA.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function h = plotcov2(mu, Sigma, varargin)

if size(Sigma) ~= [2 2], error('Sigma must be a 2 by 2 matrix'); end
if length(mu) ~= 2, error('mu must be a 2 by 1 vector'); end

[p, ...
 n, ...
 plot_opts] = process_options(varargin, 'conf', 0.9, ...
					'num-pts', 100);
h = [];
holding = ishold;
if (Sigma == zeros(2, 2))
  z = mu;
else
  % Compute the Mahalanobis radius of the ellipsoid that encloses
  % the desired probability mass.
  k = conf2mahal(p, 2);
  % The major and minor axes of the covariance ellipse are given by
  % the eigenvectors of the covariance matrix.  Their lengths (for
  % the ellipse with unit Mahalanobis radius) are given by the
  % square roots of the corresponding eigenvalues.
  if (issparse(Sigma))
    [V, D] = eigs(Sigma);
  else
    [V, D] = eig(Sigma);
  end
  % Compute the points on the surface of the ellipse.
  t = linspace(0, 2*pi, n);
  u = [cos(t); sin(t)];
  w = (k * V * sqrt(D)) * u;
  z = repmat(mu, [1 n]) + w;
  % Plot the major and minor axes.
  L = k * sqrt(diag(D));
  h = plot([mu(1); mu(1) + L(1) * V(1, 1)], ...
	   [mu(2); mu(2) + L(1) * V(2, 1)], plot_opts{:});
  hold on;
  h = [h; plot([mu(1); mu(1) + L(2) * V(1, 2)], ...
	       [mu(2); mu(2) + L(2) * V(2, 2)], plot_opts{:})];
end

h = [h; plot(z(1, :), z(2, :), plot_opts{:})];
if (~holding) hold off; end