Determining whether a set of points is contained on an ellipse

Question

My problem is quite broad, but I wanted to know if there exists any results on this: if $\mathcal{P}$ is a set of $n$ points, then what are necessary and sufficient conditions for these $n$ points being contained on an ellipse? Does there exists any theorems / lemmas on this?

Answer 1

Let $n$ be the number of points.

If $n\le 4$ and the points are not collinear, then you can always find an ellipse on which the points lie.

If $n \ge 5$, then take any $5$ of the points, and construct the conic on which these points lie. The conic equation can be found from the following determinant

$\begin{vmatrix} x^2 && xy &&y^2 && x && y && 1 \\ x_1^2 && x_1y_1 &&y_1^2 && x_1 && y_1 && 1\\ x_2^2 && x_2y_2 &&y_2^2 && x_2 && y_2 && 1\\ x_3^2 && x_3y_3 &&y_3^2 && x_3 && y_3 && 1\\ x_4^2 && x_4y_4 && y_4^2 &&x_4 && y_4 && 1\\ x_5^2 && x_5y_5 &&y_5^2 && x_5 && y_5 && 1 \end{vmatrix} = 0 $

See here for details.

If this conic is an ellipse, then you can check the other points by direct substitution in the ellipse that you found.