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

Definition of ellipse as a deformed circle

Every ellipse can be obtained by stretching some circle. Begin with any circle in the plane, and add x and y axes that place the origin at the center of the circle. Next, take every (x, y) on the circle, and change it to (2x, y). This has the effect of stretching the circle horizontally by a factor of 2, while leaving the vertical scale unchanged. See Figure 1. More generally, for any positive number r, we can use a factor of r instead of 2, changing (x, y) to (rx, y). Every ellipse can be obtained from a circle in this fashion.

Figure 1. Stretching a circle.

Degenerate Ellipse

Although the example above illustrates the case for r ≥ 1, it is equally valid to consider r < 1. In this case what we have called a stretch actually shrinks the circle along one axis. As long as r is positive, the resulting curve is a legitimate ellipse. In the limiting case of r = 0, the circle is collapsed to a line segment. This is sometimes referred to as a degenerate ellipse.

The stretch (or shrink) described above is a linear transformation and can be expressed using matrix multiplication (see Representing transformations with matrices). Representing the starting point (x, y) and its stretched image (u, v) as columns, a horizontal stretch by a factor of r is given by


Using the equation for a circle and this transformation, we can derive an equation for the ellipse. That is the next topic.