%<pre>
function a = fitellipse(X,Y)

% FITELLIPSE  Least-squares fit of ellipse to 2D points.
%        A = FITELLIPSE(X,Y) returns the parameters of the best-fit
%        ellipse to 2D points (X,Y).
%        The returned vector A contains the center, radii, and orientation
%        of the ellipse, stored as (Cx, Cy, Rx, Ry, theta_radians)
%
% Authors: Andrew Fitzgibbon, Maurizio Pilu, Bob Fisher
% Reference: "Direct Least Squares Fitting of Ellipses", IEEE T-PAMI, 1999
%
%  @Article{Fitzgibbon99,
%   author = "Fitzgibbon, A.~W.and Pilu, M. and Fisher, R.~B.",
%   title = "Direct least-squares fitting of ellipses",
%   journal = pami,
%   year = 1999,
%   volume = 21,
%   number = 5,
%   month = may,
%   pages = "476--480"
%  }
% 
% This is a more bulletproof version than that in the paper, incorporating
% scaling to reduce roundoff error, correction of behaviour when the input 
% data are on a perfect hyperbola, and returns the geometric parameters
% of the ellipse, rather than the coefficients of the quadratic form.
%
%  Example:  Run fitellipse without any arguments to get a demo
if nargin == 0
  % Create an ellipse
  t = linspace(0,2);
  
  Rx = 300;
  Ry = 200;
  Cx = 250;
  Cy = 150;
  Rotation = .4; % Radians
  
  NoiseLevel = .5; % Will add Gaussian noise of this std.dev. to points
  
  x = Rx * cos(t);
  y = Ry * sin(t);
  nx = x*cos(Rotation)-y*sin(Rotation) + Cx + randn(size(t))*NoiseLevel; 
  ny = x*sin(Rotation)+y*cos(Rotation) + Cy + randn(size(t))*NoiseLevel;
  
  % Clear figure
  clf
  % Draw it
  plot(nx,ny,'o');
  % Show the window
  figure(gcf)
  % Fit it
  params = fitellipse(nx,ny);
  % Note it may return (Rotation - pi/2) and swapped radii, this is fine.
  Given = round([Cx Cy Rx Ry Rotation*180])
  Returned = round(params.*[1 1 1 1 180])
  
  % Draw the returned ellipse
  t = linspace(0,pi*2);
  x = params(3) * cos(t);
  y = params(4) * sin(t);
  nx = x*cos(params(5))-y*sin(params(5)) + params(1); 
  ny = x*sin(params(5))+y*cos(params(5)) + params(2);
  hold on
  plot(nx,ny,'r-')
  
  return
end

% normalize data
mx = mean(X);
my = mean(Y);
sx = (max(X)-min(X))/2;
sy = (max(Y)-min(Y))/2; 

x = (X-mx)/sx;
y = (Y-my)/sy;

% Force to column vectors
x = x(:);
y = y(:);

% Build design matrix
D = [ x.*x  x.*y  y.*y  x  y  ones(size(x)) ];

% Build scatter matrix
S = D'*D;

% Build 6x6 constraint matrix
C(6,6) = 0; C(1,3) = -2; C(2,2) = 1; C(3,1) = -2;

% Solve eigensystem
if 0
  % Old way, numerically unstable if not implemented in matlab
  [gevec, geval] = eig(S,C);

  % Find the negative eigenvalue
  I = find(real(diag(geval)) < 1e-8 & ~isinf(diag(geval)));
  
  % Extract eigenvector corresponding to negative eigenvalue
  A = real(gevec(:,I));
else
  % New way, numerically stabler in C [gevec, geval] = eig(S,C);
  
  % Break into blocks
  tmpA = S(1:3,1:3); 
  tmpB = S(1:3,4:6); 
  tmpC = S(4:6,4:6); 
  tmpD = C(1:3,1:3);
  tmpE = inv(tmpC)*tmpB';
  [evec_x, eval_x] = eig(inv(tmpD) * (tmpA - tmpB*tmpE));
  
  % Find the positive (as det(tmpD) < 0) eigenvalue
  I = find(real(diag(eval_x)) < 1e-8 & ~isinf(diag(eval_x)));
  
  % Extract eigenvector corresponding to negative eigenvalue
  A = real(evec_x(:,I));
  
  % Recover the bottom half...
  evec_y = -tmpE * A;
  A = [A; evec_y];
end

% unnormalize
par = [
  A(1)*sy*sy,   ...
      A(2)*sx*sy,   ...
      A(3)*sx*sx,   ...
      -2*A(1)*sy*sy*mx - A(2)*sx*sy*my + A(4)*sx*sy*sy,   ...
      -A(2)*sx*sy*mx - 2*A(3)*sx*sx*my + A(5)*sx*sx*sy,   ...
      A(1)*sy*sy*mx*mx + A(2)*sx*sy*mx*my + A(3)*sx*sx*my*my   ...
      - A(4)*sx*sy*sy*mx - A(5)*sx*sx*sy*my   ...
      + A(6)*sx*sx*sy*sy   ...
      ]';

% Convert to geometric radii, and centers

thetarad = 0.5*atan2(par(2),par(1) - par(3));
cost = cos(thetarad);
sint = sin(thetarad);
sin_squared = sint.*sint;
cos_squared = cost.*cost;
cos_sin = sint .* cost;

Ao = par(6);
Au =   par(4) .* cost + par(5) .* sint;
Av = - par(4) .* sint + par(5) .* cost;
Auu = par(1) .* cos_squared + par(3) .* sin_squared + par(2) .* cos_sin;
Avv = par(1) .* sin_squared + par(3) .* cos_squared - par(2) .* cos_sin;

% ROTATED = [Ao Au Av Auu Avv]

tuCentre = - Au./(2.*Auu);
tvCentre = - Av./(2.*Avv);
wCentre = Ao - Auu.*tuCentre.*tuCentre - Avv.*tvCentre.*tvCentre;

uCentre = tuCentre .* cost - tvCentre .* sint;
vCentre = tuCentre .* sint + tvCentre .* cost;

Ru = -wCentre./Auu;
Rv = -wCentre./Avv;

Ru = sqrt(abs(Ru)).*sign(Ru);
Rv = sqrt(abs(Rv)).*sign(Rv);

a = [uCentre, vCentre, Ru, Rv, thetarad];