Fitting an Asphere to a set of 3D point cloud data

Hello,

I have a set of X, Y, Z values that form an "asphere", the formula for this asphere is as follows:

R = -353.5;
k = -1.0;
c = 1/R;

Form =@(x, y, k, c)  c*((x-x0).^2+(y-y0).^2) ./(1+sqrt(1-(1+k)*c^2*((x-x0).^2+(y-y0).^2))); % x0, y0 coordinates of the center of the asphere

My qustion is: how can I use MATLAB cftool (or any other way) to fit an asphere made with the above formula to the 3D data (X, Y, Z) to obtain the minimum residual.

Any help is much appreciated.

Answer 1

[darova]

I tried usual paraboloid

[Feri]

Darova,

Thank you for your response, from the fit based on your solution I get the following residual:

Which does not match the results of a software that I use for verification (that I don't know what method it is using for fitting).

What I do is that based on the asphere formula and X, Y data I create a model of the asphere, then try to fit this model to the real data, however, the model has to be rotated and translated to result in the best fit. My issue is that I can't/don't know how to move around my fit to best align it with the real data and subtract one from the other to get the minimum residual.

Could you help me with that?

Thank you very much.

[darova]

Try this script

[Feri]

Thank you drova;

I ran the script you provided and removed a plane from the residual and got this: (YZ plane view)

This is due to the decenter of the fit and the real data, I don't know what method the software is using but it doesn't have this decenter.

[darova]

I don't understand. What kind of decenter?

% c - decenter iz Z direction
% y0 - decenter in y direction
f = @(a,b,c,y0,x,y) a*x.^2 + b*(y-y0).^2 - c;

[Feri]

The data that I uploaded here is based on a theoretical model made from the equation of the asphere, the best fit should ideally result in a plane residual (with numerical errors), but this error of order 1e-3 seems to be from the offcentering of the model and the data.

Thank you.

[darova]

[Feri]

Darova I really appreciate your help but I still don’t have the answer to my question.

The residual should be a flat plane.

[darova]