Least-squares ellipse fitting

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

• Try to google "fitting the ellipse" and maybe this will help: cs.cornell.edu/cv/OtherPdf/Ellipse.pdf Aug 13, 2014 at 10:55
• Formulas look there. Or choose other. math.stackexchange.com/questions/809907/… Aug 13, 2014 at 11:15
• "I get a straight line and not really a fitted ellipse" - This doesn't sound right. What were the parameters? What values of x did you use? Can you post how you found the y values? They are not linear in x unless A,B,C are small relative to the other params, I think? Aug 13, 2021 at 11:32