Least-squares ellipse fitting

Question

I am trying to find a least-squares ellipse fit for a set of 100 data points $(x,y)$.

Now I have found the values of $A,B,C,D,E,F$ according to the conical equation of the ellipse $$ Ax^2+Bxy+Cy^2+Dx+Ey+F=0 $$ I would like to know how to find the points that actually lie on this ellipse. From my basic understanding, if I substitute a value of $x$ in the above equation, it should give me the corresponding value of $y$.

When I do the above, I get a straight line and not really a fitted ellipse. How can I find the fitted ellipse?

My task is to plot these points so that I can see the best possible fit. For reference see [link]. This is the source of ellipse fitting that I am currently using.

I appreciate help from anyone who has experience with this. I am sorry if I am lacking some basic mathematical knowledge, but from what I understand, it isn't all that straightforward.

Regards

Arj

Answer 1

A sure way to determine whether a point lies on the ellipse is to substitute the point's x- and y-coordinates into the equation and see whether the equation is exactly satisfied. (Your method is also valid, but requires more work.)

An example of how to use MATLAB to plot a curve and points is at https://www.mathworks.com/help/curvefit/fit.html. You will need to solve your equation for y. Be careful, though, because a point may appear to be on the ellipse but actually not be on it.