Caption: Ellipse diagram showing the two focuses---which are special points of an ellipse that enter into ellipse formulae and into the geometrical contruction of ellipses.
Features:
(x/a)**2+(y/b)**2=1 ,
where the origin is at the geometric center of the ellipse (i.e., the point of highest symmetry), a is the semi-major axis (i.e., half the long axis aligned with the x-axis), b is the semi-minor axis (i.e., half the short axis aligned with the y-axis), and a ≥ b without loss of generality since one can just flip the names if a < b and rotate the ellipse by 90°.
If a = b, then the ellipse specializes to the circle with radius a = b.
Proving this geometrical construction is consistent with the x-y formula above takes a bit of work. We do this in the Extended Features of the Extended File: Mathematics file: ellipse_4.html---which if this file is that file, we do the work below.
First note that usually eccentricity is given the symbol e---NOT to be confused with the exponential e.
Fiducial values of eccentricity e are: