Ellipses

The ancient Greeks knew an awful lot about a family of curves that they introduced and studied, a family called the conic sections. Of course, modern Greeks may also know a lot about these curves. These curves turn out to be basic in describing such phenomena as the orbit of a planet, let’s say. And they also explain why your satellite dish has a parabolic shape, for example. Let’s begin with an ellipse. An ellipse is described in classical terms in the following way. We’re given two different points, F1 and F2, in the plane. Let’s call the distance between them 2c. So c is a strictly positive number here.

And we let a be a number bigger than c, any fixed number bigger than c, and we now define an ellipse in the following way. We define it to be the set of points, P, which satisfy the following equation, the distance from P to F1 plus the distance from P to F2 is equal to 2a. So here’s a point P. We can calculate its distance to F1, d1. Its distance to F2. And the sum of these two distances has to equal 2a. Now, in order to describe this ellipse in more concrete terms, we need to introduce coordinate axes. You remember what our overarching theme here is? We want to make connections between geometrical objects and numbers in equations.

Now, I’ve introduced the coordinate axes in such a way that the points F1 and F2 are both on the x-axis, and the origin is exactly the midpoint between them. That’s called the canonical configuration.

And when you have this canonical configuration and you look at all the points P that are defined by the condition of an ellipse as I’ve said, then you wind up getting a curve that looks like this.

The length, the horizontal length between the origin and F1, is the number that was c. And if you look at where the x-axis is crossed, that’s one of the vertices of the ellipse. That point turns out to be the point a, 0.

If you now introduce another variable, parameter rather, b, a positive one, by means of the equation c squared equals a squared minus b squared, then the b has an interpretation on the picture as well. It turns out that the point where the y-axis is crossed, the upper y-axis, is exactly the point 0, b. So here’s our ellipse.

It’s defined by this equation. Given what we know about how Euclidean distance is defined, an equivalent way to write it is with these square roots. So here’s the equation that defined an ellipse. As you can see, it’s the distance from x, y to the point c, 0. That’s the point F1. Plus the distance from x, y to the point minus c, 0. That’s the point F2. The sum of those two distances is equal to 2a. Now, here’s an exercise for you. It’s not a trivial exercise, but I’m sure that you can do it. I ask you to derive the canonical equation of an ellipse from the one that we have currently. This one that you see.

It turns out that you can work on that equation. You can get an equivalent form of the equation. And the equivalent form turns out to be much neater. In fact, it’s this one, x squared over a squared plus y squared over b squared equals 1. Now, how do you go from the first form to the second? Well, in solving these equations with two radicals, you know that you want to put one radical on one side of the equation and the other radical on the other. Then, you square both sides of the equation that you get. There’ll still be a radical. You isolate it on one side and square again. And you’ll come to this form.

In doing this, you’ll need to remember you’ll need to use that c squared is a squared minus b squared by definition. And so this is the case where a is greater than b. And you’ll find this equation.

Now, as a approaches b in the limit, you’ll find that you get the circle as a limiting case. It’s the case in which the two points, F1 and F2, coincide with the origin. All right, so here’s our ellipse. There’s a little bit of jargon that goes with this famous conic curve. Namely, the major axis of an ellipse in this diagram refers to the horizontal segment between the two vertices, minus a, 0 and a, 0. The minor axis is the vertical segment that is determined by the ellipse. The centre of the ellipses is 0, 0 in this canonical configuration. We’ve seen what the vertices are. And the foci– foci is the plural of focus.

The foci are the two points, F1 and F2, that we started with, whose coordinates here are minus c, 0 and c, 0. There’s a beautiful geometrical fact about an ellipse which may be handy if ever you wind up playing billiards on an elliptical pool table. I grant you, this is not terribly likely. But if you have a ray that goes through the point c, 0, one of the foci, and if it then rebounds in the usual way from the boundary, the ellipse, then it proceeds to go through the other focus, the point F2. This fact has significant applications in optics, for example, or astrophysics. The equation of our ellipse.

It turns out that our first equation, as we saw, was in the case a greater than b. If b turns out to be greater than a, then it’s still an ellipse, except it’s oriented vertically rather than horizontally as indicated. If you see an equation such as written here, then you should be able to recognize it immediately as an ellipse. It’s going to be a horizontally-oriented ellipse because the 9 is bigger than the 4. And what is going to be the center of the ellipse? You should be able to read it off. It’s the point 1 minus 2. Now, of course, as with circles, the equation of ellipse might be given out in expanded form.

And you might have to come back to this canonical form by completing the square with the x-terms and also with the y-terms as we did before. You might ask, does an ellipse always have to be horizontal or vertically-oriented? Good question. No, an ellipse can be rotated or skewed. And we’ll be looking at those later on.