Do 4 points in ${\mathbb R}^2$ in convex position define a unique elliplse that passes through those 4 points? So it takes 3 distinct points in the plane, that are not collinear, to define a unique circle that passes through the points.
So what about ellipses?
Arguing naively in terms of degrees of freedom doesn't seem to help too much, because for a circle we have 3 degrees of freedom (2-d center coordinates and 1-d radius), and yet it takes 3 2-d points (6 degrees of freedom), not 2 2-d points (4 degrees of freedom), to define a unique circle that passes through the points. I haven't studied conic sections much so I apologize if this question is trivial, but if we have 4 2-d points in convex position (i.e. each point is a vertex of the convex hull) does this define a unique ellipse that passes through the 4 2-d points? Or do we sometimes need 5 or more 2-d points in convex position?

This argument is taken from How many points does it take to define...
The picture says it all:


Suppose that we have K points, and we want these points to define only a single ellipse. If the points do define two ellipses, then the points must be found on the circumference of both – so they must be on the intersections between the two ellipses. Two ellipses intersect at most at four points – so if we have only four points, we are not able to resolve the ambiguity between the two. In order for there to be a contradiction, there must be at least five distinct points. Note that this proof doesn’t say that five points are enough; it only says that four are not.

