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## Re:fitting an ellipse

**From**: |
William Lash |

**Subject**: |
Re:fitting an ellipse |

**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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