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?