Fitting data to a limaçon curve

I have data in the form of two column vectors x and y containing the x and y coordinates of measurements. The data forms a limaçon curve that can be described by the equation (x^2+y^2-a*x)^2 = b^2*(x^2+y^2). How can I fit a curve of that type to my data yielding a and b as results?

Answer 1

[Matt J]

Here's a poor man's method, assuming you don't have the Optimization or Curve Fit Toolboxes

 z=x.^2+y.^2;

 %Cheesy initial guess of p=[a,sqrt(b)]

 p= [x, sqrt(z)]\z;
 p(2)=max(p(2),0);   %b must be positive

 %Re-optimize
 p0=p;
 p=fminsearch(@(p) norm( (z-p(1).*x).^2 - z.*p(2).^2  ), p0 );

 a=p(1);
 b=p(2)^2;

[Frederik]

I do have the Optimization and Curve Fitting Toolboxes, though, but I had trouble defining fittypes for use with fit(). I also just realized there is a typo in the equation for the limaçon, it should be b^2 on the right hand side.

[Matt J]

Then you could use lsqnonlin, for example.Or, here's a modification of my code

    z=x.^2+y.^2;
    fun=@(p) norm( (z-p(1).*x).^2 - z.*p(2).^2  );

   %Cheesy initial guess of p=[a,b]

   p1 = [x, sqrt(z)]\z;
   p2 = [-x, sqrt(z)]\(-z);

   f1=fun(p1);
   f2=fun(p2);

   p0=p1*(f1<=f2) + p2*(f2<f1);

   %Re-optimize

   p=fminsearch(fun, p0 );
   a=p(1);
   b=p(2);

[Frederik]

Thanks, it worked out well!