How to fit a conic using least-square method? I have a problem and I haven´t been able to solve it. The problem is in the area of least-square fitting. Someone drew a sort of "conic" figure on a canvas (i.e. a MATLAB plot) so I have a series of points ($x_i$, $y_i$). Now I need for this points to adjust to a perfect "conic" using least-square fitting.
I now how to do this over a line $f(x) = ax + b$, but I don't know what to do with the general conic equation $Ax^2+Bxy+Cy^2+Dx+Ey+F = 0$ 
Can someone point me into the right direction.
Some remarks I've been working around:


*

*that is an equation, not a function, do i need to parametrize it? how to?

*Don't know what to do with rectangle term ($Bxy$)


Thanks in advance.
PS: If there was something unclear, please say so and I'll try to explain myself.
PS.1: MATLAB code will be appreciated
Edit:
After checking what @ClaudeLeibovici suggested I got working an example on MATLAB where after getting the points I solve the system of equations given in the french paper and It plotted a perfet circle as I needed (using ezplot with the "explicit equation"), but it does not satisfies Hyperbolas or Parabolas, just circles and elipses. And i need to fit any conic.
Sometimes I get what it looks like to be an hyperbola, sometimes not in the right direction and sometimes not even close to the one I "drew".
Anyhow, I will apreciate if someone could explain me where did the "Generalization for conics" from the paper came from? and what does it means in a least-square sense?
I tried reading it, but couldn't understand it very well, cause I do not speak nor understand french, and the translation wasn't good. I need to understand what is going on cause I need to do a presentation about this.
Thanks in advance
PS: I used $F=1$ as suggested.
 A: The basic idea is to minimize $$\Phi=\sum_{i=1}^n (Ax_i^2+Bx_iy_i+Cy_i^2+Dx_i+Ey_i+F)^2$$ Take the derivatives with respect to each parameter and set it equal to $0$.
If you are lazy, define $z_i=0$ for all $i$'s and perform a least square fit for $$z=Ax^2+Bxy+Cy^2+Dx+Ey+F$$ Just a multilinear regression then.
Warning :   As JeanMarie commented, set $F=1$ or whatever number you want. This fix to $5$ the number of parameters to be adjusted. And $5$ is the minimum number of points which define a general conic not going through the origin.
A: I will assume that
this drawing is closed,
so that
you have a set of points
that you want to fit
an ellipse to
and that these points
are approximately equally spaced.
First,
get the average
$x$ and $y$ values
as $x_0$ and $y_0$.
This shows the center of the ellipse.
Subtract $(x_0, y_0)$
 from each point.
Next,
do an orientation-independent
linear least squares fit
to the modified data points.
Here is a link to my method
of doing this:
linear least squares minimizing distance from points to rays - is it possible?
For each angle $\theta$
in the expression
$D=D_1+R\cos(2\theta-\phi)$,
where $R$ and $\phi$
are specified in my answer,
the value of $D$
is the
sum of the squares
of the errors
of the line at angle $\theta$.
By choosing $\theta$ so
$\cos(2\theta-\phi) = -1$
(i.e. $\theta = (\phi+\pi)/2$),
the sum is minimized,
and by choosing $\theta$ so
$\cos(2\theta-\phi) = 1$
(i.e. $\theta = (\phi-\pi)/2$),
the sum is maximized.
These values,
$D_1 \pm R$,
give the square of the values of the
major and minor axes
of the ellipse,
scaled by $\sqrt{2}$
(this is my vague recollection - 
I did this over thirty years ago).
This method is linear
and gives you the
center,
major axis,
minor axis,
and orientation of
the major axis of the ellipse.
You may have to
play around with it
to scale and select
the length of the axes,
but it should work.
