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 is an image to accompany a point I was making in a comment:

Four points ---here, the cyan ones at the corners of a square near the center--- don't just determine infinitely-many ellipses; they determine infinitely-many hyperbolas, too. (No parabolas this time.) Moreover, those ellipses and hyperbolas cover disjoint regions of the plane, in such a way that each point on the plane (other than the first four) lies on exactly one curve. This is what it means for a fifth point added to the collection to determine one of those curves uniquely. This is also why there has to be a "special" condition (namely: in which region to place the point) to guarantee that the fifth point determines the specific type of curve (ellipse or hyperbola) you might want. 
