I've got a set of (r,z) data which represent a surface section. I would like to fit such data to the Even asphere expression:
Where 
is the surface sagita, r is the radial coordinate, R is the radius of curvatre at the vertex, κ is the conic constant and 
are the coefficients describing the deviation of the surface from a pure conic section. Can anyone help to address the problem? I have calculated the value of R by first fitting the data to a sphere with the code attached below, but I can't find a way to fit the data to the term 
once R has been calculated. Thank you, your help is very much appreciated!
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Function to fit circle to data set:
function Par = CircleFitting(XY)
centroid = mean(XY); % the centroid of the data set
X = XY(:,1) - centroid(1); % centering data
Y = XY(:,2) - centroid(2); % centering data
Z = X.*X + Y.*Y;
Zmean = mean(Z);
Z0 = (Z-Zmean)/(2*sqrt(Zmean));
ZXY = [Z0 X Y];
[U,S,V]=svd(ZXY,0);
A = V(:,3);
A(1) = A(1)/(2*sqrt(Zmean));
A = [A ; -Zmean*A(1)];
Par = [-(A(2:3))'/A(1)/2+centroid , sqrt(A(2)*A(2)+A(3)*A(3)-4*A(1)*A(4))/abs(A(1))/2];
end
