Now that you have shown the problem you are trying to solve...
I'm pretty sure I remember telling you that you should not use the function I posted. :) Sadly, it looks like you managed to find polyfitn, but did not believe me. While I admit that the reason I suggested you would not want to use polyfitn was because I was guessing you would try to fit too high order of a polynomial to some cloud of points, you still cannot use polyfitn here.
You have shown what is apparently a set of points on a sphere.
So, suppose that I gave you a simple equation of a sphere? It might look like this:
Although a classic form that would allow a non-zero center and some more general radius would look like this:
(x-x0)^2 + (y-y0)^2 + (z-z0)^2 = R^2
In either case, are these things a polynomial function of x? Are they a polynomial at all? Actually, no. They are not so, in either case. They look kind of polynomial-ish. But in fact, they are equations, describing what is technically called an implicit function . Now, you might want to write them in the form
z(x,y) = +/- sqrt(2 - x^2 - y^2)
but even that is not a function. That little +/- in the beginning should be the clue that a problem exists, that this is not a single valued animal, so you cannot think of it as z(x,y).
A single valued function would have the form z(x,y). For any value of x and y that you would put in, you get ONE value of the function out. The polyfitn tool is designed to work with (estimate) polynomials. It applies to single-valued functions of one or more independent variables.
So, what do you have? You have a more difficult thing that is not in fact a single valued function at all. You have what might be also correctly described as a multivalued function. Really, it is the surface of what looks to be a sphere. I might guess this is a test case, that your real problem is not quite so simple, and you really have some more nasty looking 2-manifold embedded in R^3 that you want to fit as a polynomial. Or it may be really that simple, and you just want to find the equation of a sphere from some data points. But you cannot use polyfitn to fit an implicit/multi-valued function. polyfitn assumes that you have data that represents a single valued function, here z(x,y). Pass in (x,y), get one z out. You don't have that.
You cannot use polyfitn. You cannot use the curve fitting toolbox. You cannot use regress. You cannot use scatteredInterpolant. You cannot use griddata. You cannot use my other tools, in case you chance upon gridfit.
What not? Again, you don't have data in the form of a single valued function. The surface of a sphere is not a function. It is defined by an implicit functional form, thus one of those I showed above.
What happens when you try to use a tool like polyfitn to data that lives on the surface of a sphere? You might get some simple plane, that passes through the center of the sphere, or you might get something more nasty looking that still fails for many reasons.
Now, maybe what you really want to do is just find the equation of a sphere from some data in three dimensions? You could look on the File Exchange, which is what I should probably tell you to do. There is a lot of crap on the FEX for things like this though. So I would need to find something on the FEX that does work, or I could just be lazy and attach circlefit.m to this answer. I did the latter. It finds the center and radius for a circle (or sphere, or hypersphere), fit to n-dimensional data. I wrote it many years go when playing around with some ideas for how I might robustly solve the problem, but it should still work.
To be honest, I'm not positive that in the end your real problem really is as simple as fitting a sphere to some data. It never is. :) In fact, I worry that your problem is something more general, because you talk about interpolating the surface. People show a sphere, but they will really want to fit the surface of some amorphous non-convex blob when you see their real data. So if it just fitting a sphere, then circlefit.m will do it simply enough. If it is the amorphous blob thingy, then you need to say so. I never got around to writing an amorphousblobfit.m though.
