Ellipses

Recall: An ellipse is a geometric shape that is defined by the a set of points, such that the sum of the distance between each of these points and two foci is constant. It can also be defined as the intersection between a plane and a hollow cone. The eccentricity of an ellipse is defined as the ratio between the distance between the two foci, and the length of its major axis. An ellipse must have a positive eccentricity less that 1, and if it has an eccentricity of 0, it is a circle.

The general form of an ellipse is: \begin{align*}\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\end{align*}

You can easily visualize an ellipse in real life. For example, if you tilt a circular object at an angle, you can observe the an ellipse. Did you know that planets also move in elliptical orbits around the sun? This is because the sun’s gravity and the planet’s velocity cause the planet to curve around the sun. The sun is one of the foci of the ellipse and the other is simply a random point in space. Some objects have very low eccentricities, like the earth, and some have high eccentricities, like comets.

Whispering Galleries

What’s that? The National Statuary Hall in Washington, D.C. is in the shape of an ellipse! Rooms in the shape of ellipses are known as whispering galleries because you can literally hear someone’s whisper from across the room! To do this, you must stand at the one of the foci and have a friend (or a stranger!) stand at the other foci in the room. You will be able to communicate to each other from across the room without anyone else hearing you! That is, unless, they are standing right behind you... Whispering galleries work because your sound waves, from one focus point, bounce off the curves of the elliptical wall and pass through the other focus point.

Creative applications

1. Why can’t an ellipse have an eccentricity greater than 1?

2. What else is in the shape of an ellipse?

3. A circle is an ellipse where the foci are ____?