Variation Analysis, Ellipse

central equation:

For the ellipse we have four different parameters a, b, x, and y, whose changes we'll study for their effects on the graph of the ellipse.

Shown is an ellipse whose major axis is along the X axis. (What follows holds with the vertical ellipse whose major axis is along the Y axis.)

C is the distance to a focus from the origin. a and b are the lengths of the semi major and semi minor axis, respectively. (A vertical ellipse would have b as the length of the semi major axis since it lies on the Y axis, and a as the length of the semi minor axis along the X axis.)

Say we keep a constant, that is the length of the semi major axis constant. Now lets vary c. Remember, a is constant, so the left hand side of the following equation does not change.

a² = c² ↑ + b²↓

and

a² = c² ↓+ b² ↑

The triangle shows this relationship. If c increases and a stays constant then the hypotenuse a must rotate toward the leg c. For this to occur, the other leg b must decrease.

Herea = 6. We start off with the cyan ellipse. Now move the focal points toward the origin by making c smaller and smaller. b climbs the Y axis to the limiting point of b = a. At his point our ellipse has become a circle with radius a=b. c² decreases to 0 as c decreases to 0, so we then have

a² = c² ↓ ₀ + b² ↑             (↓₀ means decreases toward 0)

until a² = 0 + b²

Now b = a and c is the origin, which gives us a circle (the magenta ellipse).

If instead, we start with c = 1, the magenta ellipse and increase it to c = 5.7, then the ellipse flattens to the cyan ellipse. If c continues to increase until c = 6, that is, c = a, then our ellipse has degenerated into two line segments, one for the top and one for the bottom of the ellipse from x = -6 to x = 6.

a² = c² ↑ + b² ↓

until a² = c²

which means a² - c² = 0 = b² so b = 0

Now let's keep the focal points constant, i.e., c constant and vary a (or b).

If a increases, so must b to maintain the equality. (Likewise if a decreases so must b.)

   a²↑ = c² + b²↑

and

   a²↓ = c² + b²↓

We start with the green ellipse. The two black dots are the foci. Now as we increase a, b also increases giving the cyan ellipse. When we increase a to its final value of 8, b has increased to 5.7 and we get the purple ellipse.

Let's have another look at a² = c² + b²

If we allow a to continue increasing then, with c constant, b must increase as well. The length of the minor axis will approach the length of the major axis (but never never become equal) but, becoming so large that c will be insignificant and the ellipse approaches a very large circle whose radius is a.

(Since a and b may continue to increase with no bound, c will continue to decrease towards 0 which is its bound. Not until a = b will we have a circle. But a will never exactly equal b, so c will never exactly become 0.)

a² ↑↑ = c² ↓↓ + b² ↑↑

Now suppose we make a equal to b. If we allow b to increase then the major axis moves form the X axis to the Y axis and as b continues to increase we start seeing a vertical ellipse whose foci move away from the origin along the Y axis.

When talking about an ellipse qualitatively (and later with the hyperbola) we usually talk about the ratio

This ratio is called the eccentricity of the conic and it's customary to use the letter e to name this ratio. Euler's constant is 2.71... and the same letter e is used to name that constant. This is fine since you know which context it is being used. Here, however, to emphasize the difference, I'll use the lower case Greek letter epsilon ε.

                                   ε =

As indicated previously, a, b, and c form a right triangle with a as the hypotenuse. a, b, and c cannot become 0 if we have an ellipse and a must always be greater than c (the hypotenuse of a right triangle is always greater then either of its legs.) Also, for a vertical ellipse remember that b plays the role of a so, in this case, ε = c/b_.)

since neither a nor c can be zero in a right triangle

now combining the inequalities

So, for an ellipse we have:

                            0 < ε < 1

We've just shown that as ε → 1 the ellipse flattens and

as ε → 0, the ellipse approaches a circle.

From the smallest to the largest eccentricity we have the red ellipse #1, then the blue ellipse #3, and finally the gray ellipse #2.

Here we see three ellipses that have the same eccentricity. The triangles show the relationships between a, b, and c for each ellipse. These right triangles are similar triangles; the ratios of their sides are the same. In particular c/a is the same.

ADVANCED: (ALGEBRA 2)

There's a deeper meaning in the eccentricity. And this meaning comes from its derivation which follows. The algebra is a bit involved but is explained fully.

An ellipse has a directrix, but it hasn't been necessary to discuss until now. What we want to show is that for any point on the ellipse, the ratio of the distance from that point to a focus (d₁) and the distance from that point to the directrix ( d₂) is constant and equals ε. These distances are shown above.

We want to show that d₁/d₂ = ε.

The strategy we'll use is to find expressions in terms of the constants a and c and relate them to the common point (x,y). We'll see that by doing this we eliminate the x and y.

last equation simplified

definition of an ellipse, will be used now and again shortly (4 steps later)

divide the previous two equations noting that r² - d₁² = (r + d₁)(r - d₁) so

(r + d₁)(r - d₁) / (r + d₁) = (r - d₁)

now solve for d₁

We need to eliminate r so that we can use d₁ alone in terms of a, c and x. Use the previous red equation above to replace r.

Now, solve for d₁, then for x*.

From the graph: the distance from the directrix to P(x,y) is the distance from origin to the directrix (a²/c) minus the distance from the origin to x, which is x. ( referring to the x in the point P(x,y)) Substitute this expression for x into the next equation.

Doing this we eliminate x.

Simplify, watching the minus sign in front of the a.

Multiply both sides by c, then divide both sides by ad₂

like this:

cd₂ = ad₁

cd₂ / ad₂ = ad₁/ad₂

c/a = d₁/d₂

* Here are the missing steps:

d₁ = a - cx/a

d₁ - a = -cx/a

a(d₁ - a) = -cx

a(d₁ - a)/(-c) = -cx/(-c)

a(a - d₁)/c = x

subtract a from both sides

divide both sides by -c

(dividing by a negative number interchanges the terms in a difference)