We now move to the algebraic side of ellipses. The standard equation of an ellipse centered at the origin [the point (0,0)] is given by
where c and d are some positive constants. The shape of a given ellipse will depend on the relationship between c and d, as will be noted in the examples below.
Some other terms need to be introduced at this point of the discussion. The line through the foci intersects the ellipse at two points, known as vertices. (Vertices is the plural of the term vertex.) This line segment joining the vertices is called the major axis and its midpoint is called the center of the ellipse. The minor axis is the line segment perpendicular to the major axis which also goes through the center and touches the ellipse at two points. (By the way, these two points do not appear to have special names in most of the mathematical literature.) See FIGURE E3 for a graphical view of some of these key terms.
Given these terms, we need to “refine” our discussion of the standard equation of an ellipse. We say that the standard equation of an ellipse centered at the origin is given by
if the major axis is horizontal, or
if the major axis is vertical. This will insure that we always have the relationship
We can then also note that the distance c of the foci from the center of the ellipse can be determined using a, b, and the following equality:
We will use this relationship often, so keep it in mind.
The next question you might ask is this: “What happens to the equation
if the center of the ellipse is not 
?” That is an excellent
question. It turns out that the modification in the equation to accomodate
this is not difficult to make. If the center of the ellipse is
, then
the equation of the ellipse becomes
if the major axis is horizontal, or
if the major axis is vertical.
One last term related to ellipses should be noted before moving on. This is the concept of the eccentricity of an ellipse. The eccentricity e of an ellipse is defined to be the ratio of the distance between the foci to the distance between the vertices. Hence,
Note that if the foci are close to the vertices, then the eccentricity will be close to 1. (This is visually interpreted as having a fairly flat ellipse.) Note also that e can never be greater than 1 since the foci are always inside the ellipse. This means that the distance between the foci must always be less than the distance between the vertices. Thus, our fraction
will always be less than 1.
What if the foci are very close to one another? Then, c will be quite small and the eccentricity will be very close to 0. Does the eccentricity of an ellipse ever equal 0? The answer is “yes”. Remember that a circle is a special case of an ellipse in which, if you will, the two foci have become one. In this case the value of c is 0 (because the distance between the two foci is 0) and this makes
Visually, an eccentricity close to 0 then results in a fairly round ellipse.
I include FIGURE 3-5 here to show three different ellipses with the same vertices but different eccentricities.
Can you determine which has an eccentricity of 0?
Given all of this theoretical discussion, let's move to some examples.
James A. Sellers