The Journal of Online Mathematics and Its Applications, Volume 8 (2008)
The Most Marvelous Theorem in Mathematics, Dan Kalman

Overview concepts of ellipses and conic sections

An ellipse is a circle that has been stretched in one direction, to give it the shape of an oval. But not every oval is an ellipse, as shown in Figure 1, below. There is a specific kind of stretching that turns a circle into an ellipse, as we shall see on the next page. Figure 2 hints at the nature of the type of stretching that creates an ellipse. Note that a typical point of the red circle is shown, as well as the corresponding point of the blue ellipse. The blue point is the result of stretching the red point horizontally by a factor of 2.

Figure 1. Ovals vs Ellipses.

Figure 2. Stretching a circle.

There are several alternate ways to define an ellipse. One is as the intersection of a plane with a cone. An ellipse is just one of the possible shapes for this intersection (called a conic section), as shown in the Figure 3 below, which appears at Wolfram Math World

Figure 3. Sections of a cone

Sections of a cone

A second alternate definition is given in terms of two special points, called foci. A point is on the ellipse if and only if the distances from that point to the foci, when added together, result in a predefined constant. For a more detailed look at this idea, continue through three more pages in this outline, or jump ahead immediately to page 4.

Yet another way to define an ellipse involves a focus and a line called a directrix. In this definition the condition for a point to be on the ellipse is that the ratio of the distances from the point to the focus and from the point to the directrix assumes a particular value.

How do we know that all of these definitions for ellipses are equivalent? That is, how do we know that a curve that satisfies one of the definitions must necessarily satisfy the others? One approach is to use analytic geometry, developing algebraic formulations of the different definitions and seeing that they are consistent. That is the approach we will follow in this article. However, there are also other methods for establishing the equivalence of different definitions of ellipses. A beautiful geometric construction shows that an oval produced by cutting a cone with a plane satisfies the definition of an ellipse featuring two foci. See http://mathworld.wolfram.com/DandelinSpheres.html.

Ellipses (and more generally, conic sections) have an extremely rich set of properties and attributes. Some of these are more readily established using one definition, and while others benefit from using a different definition. That is why so many different definitions are perpetuated. For our purposes, we will focus primarily on just two definitions, involving deformed circles in the first case and the two focus formulation in the second. As a first step, we will begin with the definition of an ellipse as a particular kind of deformed circle. For much more information about ellipses, see http://mathworld.wolfram.com/Ellipse.html.