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$1$-ellipse is the circle with $1$ focus point, $2$-ellipse is an ellipse with $2$ foci. $n$-ellipse is the locus of all points of the plane whose sum of distances to the $n$ foci is a constant.

I know that $3$ non-collinear points determine a circle. $5$ non-collinear points on a plane determine an ellipse.

After that my question is: how many non-collinear points determine an $n$-ellipse on a plane?

Futhermore: is there a unique shape which is kind of generalization of circle or ellipse and it is determined by $4$ given non-collinear points on a plane? What can we say in this case? Is there a special fitted unique closed curve for any points?

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The number of points needed to identify a $n$-ellipse is $2n+1$. This directly follows from the general equation of a $n$-ellipse

$$\sum_{i=1}^n \sqrt{(x-u_i)^2+(y-v_i)^2}=k$$

where the number of parameters is $2n+1$. So, for a $1$-ellipse (circle) we need $3$ noncollinear points to identify $3$ parameters ($u_1,v_1,k$), for a $2$-ellipse we need $5$ noncollinear points to identify $5$ parameters ($u_1,v_1,u_2,v_2,k$), and so on.

As regards the "shape" identified by $4$ points, since these points allow to define a $2$-ellipse with the exception of a single parameter that remains unknown, the resulting figure is a set of $2$-ellipses. For example, we could use the $4$ points to calculate $u_1,v_1,u_2,v_2$, leaving $k$ unknown. This would create a set of $2$-ellipses where the only variable parameter is $k$, that is to say the sum of the distances to the two foci.

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