The Journal of Online Mathematics and Its Applications, Volume 8 (2008)

The Most Marvelous Theorem in Mathematics, Dan Kalman

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 (2`x, 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.

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

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