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Vector Graphics
No, two points define a line. It takes three points to define a plane.
A defining attribute is that it has two pairs of parallel sides. There are lots of other attributes which can be derived from this.
Four sides of equal length in two parallel pairs as well as other attributes that can be derived from these.
The two fixed points are the foci but these do not define the shape of the ellipse. You also need to know the eccentricity.