RWA# 1: Unit M Concepts 4-6: Conic Sections in Real Life

1. Definition
An ellipse is the set of all points such that the sum of two points called foci equals a constant.

2. Description
The equation of an ellipse is as follows:

(x-h)^2 + (y-k)^2

   a^2          b^2

The x and y variables represent the coordinates of a point on the graph of the ellipse. The h and k variables represent the x value and the y value of the center, respectively. The a and b values determine the distance of the side of the ellipse from the center along the major and minor axes, respectively. The length of the major axis is double the value of a, and the length of the minor axis is double the value of b. To differentiate between a and b, you must keep in mind that a will always be the larger number, regardless of whether it is under the y term or the x term.

The graph of an ellipse is as follows:

http://www.csgnetwork.com/ellipse.png

As you can see, there are more variables that cannot be known just by looking at the equation. The c value is the distance from the center to the foci. It is always located along the major axis. To find c, use the equation a^2 - b^2 = c^2. The foci are two points that are also along the major axis. When you connect the foci to another point on the ellipse (see the blue), the sum will always be a constant. Additionally, the closer the foci are to the center, the more circular the ellipse will be. This value of how much the conic section deviates from being circular is called eccentricity. To find eccentricity, use the equation c/a.

3. Real World Application
One special property of an ellipse is that energy or sound directed at one focus will reflect to the other focus. A real world application of this property is called a whispering gallery. One person can stand at the focus of the whispering gallery and say something. However, whatever they say will not be heard by anyone except someone who is at the other focus. The sound waves bounce off the sides of the ellipse and reflect to the other focus.

A video of how the foci of an ellipse are related is as follows:

As the video shows, the sum of the distance between the two foci and the two points stays constant throughout the formation of the ellipse. A sound wave travelling a certain distance through one of the foci and bouncing off the side of the wall is like making the first segment. Since it can only go a certain distance more, it must reflect toward and through the other focus.

4. Works Cited
Image of an ellipse found here.
Definition of a whispering gallery found here.
Video link for the construction of an ellipse found here.