# How to set a custom equation to fit 5 points in space by fitsurface?

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laura bagnale on 28 Jul 2021
Commented: laura bagnale on 3 Aug 2021
Hello everyone,
it would be very nice if someone could help me with this problem.
I have a set of 5 points in space with x, y, z coordinates and I would like to fit them with a surface. Generally I use fitsurf setting the option 'poly22' to get the equation in the secondo order of x and y.
Now I would like to fit them with the equation f(x,y) = ax + by + cxy + d . Is there a way to do it? Can we set the custom equation using fitsurface?
Regards,
Laura

Matt J on 2 Aug 2021
Another way, which would allow you stay within the framework of the Curve Fitting Toolbox, would be to do a poly22 fit with upper and lower bounds,
lb=-inf(1,6); lb([4,6])=0; ub=-lb; fitsurface=fit([x,y],z, 'poly22','Lower',lb,'Upper',ub)
Linear model Poly22:
fitsurface(x,y) = p00 + p10*x + p01*y + p20*x^2 + p11*x*y + p02*y^2
Coefficients (with 95% confidence bounds):
p00 = 1.319 (1.215, 1.423)
p10 = -0.0002893 (-0.0003375, -0.0002411)
p01 = -1.19 (-1.37, -1.01)
p20 = 0 (fixed at bound)
p11 = 0.0002666 (0.0001826, 0.0003507)
p02 = 0 (fixed at bound)
laura bagnale on 3 Aug 2021
Ok thank you very much for your help and for the explanation!
Kind Regards,
Laura

Matt J on 28 Jul 2021
Edited: Matt J on 28 Jul 2021
I can't find "fitsurface" in the Mathworks documentation, but the fit is easy enough to do algebraically.
x=x(:); y=y(:); z=z(:); %ensure column vectors
params=num2cell([x,y,x.*y,x.^0]\z);
[a,b,c,d]=deal(params{:})
Matt J on 2 Aug 2021
One you have the coefficients, it is eay to plot the surface.
x=x(:); y=y(:); z=z(:); %ensure column vectors
params=num2cell([x,y,x.*y,x.^0]\z);
[a,b,c,d]=deal(params{:});
fz=@(x,y) ax + by + cxy + d;
fsurf(fz);

R2021a

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