12.6 Ellipses

Only two shapes left! But these two are doozies. Expect to spend at least a couple of days on each—they get a major test all to themselves.

In terms of teaching order, both shapes are going to follow the same pattern that we set with parabolas. First, the geometry. Then, the machinery. And finally, at the end, the connection between the two.

So, as always, don’t start by telling them the shape. Let them do the assignment “Distance to this point plus distance to that point is constant” in groups, and help them out until they get the shape themselves. A good hint is that there are two pretty easy points to find on the $x$ -axis, and two harder points to find on the $y$ -axis. As always, keep wandering and hinting until most groups have drawn something like an ellipse. Then you lecture.

The lecture starts by pointing out what we have. We have two points, called the foci. (One “focus,” two “foci.”) They are the defining points of the ellipse, but they are not part of the ellipse. And we also have a distance, which is part of the definition.

Because the foci were horizontally across from each other, we have a horizontal ellipse. If they were vertically lined up, we would have a vertical ellipse. You can also do diagonal ellipses, but we’re not going to do that here.

Let’s talk more about the geometry. One way you can draw a circle is to thumbtack a piece of string to a piece of cardboard, and tie the other end of the string to a pen. Keeping the string taut, you pull all the way around, and you end up with a circle. Note how you are using the geometric definition of a circle, to draw one: the thumbtack is the center, and the piece of string is the radius.

Now that we have our geometric definition of an ellipse, can anyone think of a way to draw one of those? (probably not) Here’s what you do. Take a piece of string, and thumbtack both ends down in a piece of cardboard, so that the string is not taut. Then, using your pen, pull the string taut.

Now, pull the pen around, keeping the string taut. You see what this does? While the string is taut, the distance from the pen to the left thumbtack, plus the distance from the pen to the right thumbtack, is always a constant —namely, the length of the string. So this gives you an ellipse. I think most people can picture this if they close their eyes. Sometimes I assign them to do this at home.

OK, so, what good are ellipses? The best example I have is orbits. The Earth, for instance, is traveling in an ellipse, with the sun at one of the two foci. The moon’s orbit around the Earth, or even a satellite’s orbit around the Earth, are all ellipses.

Another cool ellipse thing, which a lot of people have seen in a museum, is that if you are in an elliptical room, and one person stands at each focus, you can hear each other whisper. Just as a parabola collects all incoming parallel lines at the focus, an ellipse bounces everything from one focus straight to the other focus.

OK, on to the machinery. Here is the equation for a horizontal ellipse, centered at the origin.