From help-octave-request at bevo dot che dot wisc dot edu Sat Dec 21 15:39:08 2002
Subject: Re:fitting an ellipse
From: "William Lash" <zlash at mail dot notwires dot com>
To: <help-octave at bevo dot che dot wisc dot edu>
Date: Sat, 21 Dec 2002 15:38:04 -0600

It's been a while since I have done any of this, but you could
use the pinv() function to do the fit.  I don't remember if
the psuedo-inverse is equivalent to a least squares fit, but
it will provide a fit for this case.

I would rewrite your equations as:

z=A*u

where the first row of z is x and the second is y, A is
the matrix [ a 0; 0 b], and the fist row of u is sin(t)
and the second is cos(t), e.g. if a is 2 and b is 1:

t=[0:0.01:2*pi];
A=[2 0;0 1];
u=[sin(t);cos(t)];
z=A*u;
plot(z(1,:)',z(2,:)');

Now to get A from z and u, you can multiply each side
by the psuedo-inverse of u, leaving you with

Aest=z*pinv(u)

Which in the case here, returns:

Aest =

    2.0000e+00    3.4534e-16
   -1.7144e-18    1.0000e+00

You could add some noise to z, since the observations will have
some noise:

zobs=z+0.2*randn(size(z));

and do the psuedo-inverse again:

 Aest=zobs*pinv(u)
Aest =

  2.0138864  0.0061711
  0.0089173  1.0124481

and then calculate the new data:


z2=Aest*u;
plot(z2(1,:)',z2(2,:)');


I guess the problem here is that you are not limiting the fit to
x=a*sin(t), but instead to x=a*sin(t)+a1*cos(t), and similarly for
y, so you may want to continue to look for something better.

Bill
 
                 



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