function Par = CircleFitByPratt(XY)

%--------------------------------------------------------------------------
%  
%     Circle fit by Pratt
%      V. Pratt, "Direct least-squares fitting of algebraic surfaces",
%      Computer Graphics, Vol. 21, pages 145-152 (1987)
%
%     Input:  XY(n,2) is the array of coordinates of n points x(i)=XY(i,1), y(i)=XY(i,2)
%
%     Output: Par = [a b R] is the fitting circle:
%                           center (a,b) and radius R
%
%     Note: this fit does not use built-in matrix functions (except "mean"),
%           so it can be easily programmed in any programming language
%
%--------------------------------------------------------------------------

n = size(XY,1);      % number of data points

centroid = mean(XY);   % the centroid of the data set

%     computing moments (note: all moments will be normed, i.e. divided by n)

Mxx=0; Myy=0; Mxy=0; Mxz=0; Myz=0; Mzz=0;

for i=1:n
    Xi = XY(i,1) - centroid(1);  %  centering data
    Yi = XY(i,2) - centroid(2);  %  centering data
    Zi = Xi*Xi + Yi*Yi;
    Mxy = Mxy + Xi*Yi;
    Mxx = Mxx + Xi*Xi;
    Myy = Myy + Yi*Yi;
    Mxz = Mxz + Xi*Zi;
    Myz = Myz + Yi*Zi;
    Mzz = Mzz + Zi*Zi;
end
   
Mxx = Mxx/n;
Myy = Myy/n;
Mxy = Mxy/n;
Mxz = Mxz/n;
Myz = Myz/n;
Mzz = Mzz/n;

%    computing the coefficients of the characteristic polynomial

Mz = Mxx + Myy;
Cov_xy = Mxx*Myy - Mxy*Mxy;
Mxz2 = Mxz*Mxz;
Myz2 = Myz*Myz;

A2 = 4*Cov_xy - 3*Mz*Mz - Mzz;
A1 = Mzz*Mz + 4*Cov_xy*Mz - Mxz2 - Myz2 - Mz*Mz*Mz;
A0 = Mxz2*Myy + Myz2*Mxx - Mzz*Cov_xy - 2*Mxz*Myz*Mxy + Mz*Mz*Cov_xy;
A22 = A2 + A2;

epsilon=1e-12; 
ynew=1e+20;
IterMax=20;
xnew = 0;

%    Newton's method starting at x=0

for iter=1:IterMax
    yold = ynew;
    ynew = A0 + xnew*(A1 + xnew*(A2 + 4.*xnew*xnew));
    if (abs(ynew)>abs(yold))
        disp('Newton-Pratt goes wrong direction: |ynew| > |yold|');
        xnew = 0;
        break;
    end
    Dy = A1 + xnew*(A22 + 16*xnew*xnew);
    xold = xnew;
    xnew = xold - ynew/Dy;
    if (abs((xnew-xold)/xnew) < epsilon), break, end
    if (iter >= IterMax)
        disp('Newton-Pratt will not converge');
        xnew = 0;
    end
    if (xnew<0.)
        fprintf(1,'Newton-Pratt negative root:  x=%f\n',xnew);
        xnew = 0;
    end
end

%    computing the circle parameters

DET = xnew*xnew - xnew*Mz + Cov_xy;
Center = [Mxz*(Myy-xnew)-Myz*Mxy , Myz*(Mxx-xnew)-Mxz*Mxy]/DET/2;

Par = [Center+centroid , sqrt(Center*Center'+Mz+2*xnew)];

end    %    CircleFitByPratt