Surface fit in polar coordinates

Hello,

I've looked elsewhere in the formum and although there are close topics I haven't quite found what I'm looking for.

I have a set of data matrices X,Y,Z. X,Y are a regular lattice of Cartesian coordinates, Z is a quantity of which I want to study the dependence on X and Y.

My data looks like in this example plot (contourf):

I want to fit these data to an antisymmetric function of the type

or symilar function, so that I'm able to remove the antisymmetric part from the data and guess/find higher harmonic contribution in ϕ, for example.

I know some fitting methods in matlab, but what would you use to best solve this case and why?

Thanks in advance!

[David Goodmanson]

Hi Alessio,

I am going to assume an odd number n of array points for x and y, with x=y=0 right in the middle, with maybe some appropriate cropping of the figure above. Since cos(phi) is odd in x and even in y, you can at least get an initial idea by taking a look at

n = size(x,2);
n2 = (n+1)/2
x1 = x(:,n2:end);
y1 = y(:,n2:end);
z1 = z(:,n2:end) + z(:,n2:-1:1);
figure(1)
contourf(x1,y1,z1); colorbar

n = size(x,1);
n2 = (n+1)/2
x1 = x(n2:end,:);
y1 = y(n2:end,:);
z1 = z(n2:end,:) - z(n2:-1:1,:);
figure(2)
contourf(x1,y1,z1); colorbar

each of which should approximately remove the large cos dependence and give an idea of what is left over. (it removes a few higher order angle dependencies too, though). I included colorbar since it shows the size of the result. If n is even, you can either rework the expressions or crop off an appropriate row and/or column

[Alessio]

Thanks for your comment.

Since X and Y are the coordinates for a triangular lattice it may be a bit tricky to crop them to render them symmetric and apply your suggestion, but I'll try. I'm afraid though that whatever extra contribution I'm looking for might cancel out as well in this way.

Best

[David Goodmanson]

aha, I didn't realize you have a triangular lattice, which changes things

Answer 1

[KSSV]

Read about cart2pol ......you can convert your X, Y into r and phi...using this. LAter you can substitute this r and phi, into your function.

[Alessio]

Hi KSSV, thanks for your comment.

Actually this was my first approach:

Convert X,Y to polar:

phi = atan2(Y,X);
r = sqrt(X.^2+Y.^2);

then try to fit with a custom function:

f = @(a1,a2,pp,rr) a1 * cos(pp) * rr * exp(-a2*rr.^2);
[phiFit,rFit,zFit] = prepareSurfaceData(phi,r,z);
fz = fit( [phiFit, rFit], zFit, f, ...
    'StartPoint', [0.0025, 0.01], ...
    'Lower', [0, 0], ...
    'Robust', 'LAR' )

I keep getting this error when trying to fit:

Error using fittype>iDeduceCoefficients (line 619)
The independent variable x does not appear in the equation expression.
Use x in the expression or indicate another variable as the independent variable.

I'm not sure how to specify the independent variable and I'm also not sure I'm doing everything correct.

Any feedback?

I've just stumpled upon THIS that uses nlinfit, I'll try to look also in that direction.

Best