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

Linear transformations take ellipses to ellipses

The proof of Marden's Theorem depends in various places on the ability to apply geometric transformations to figures without disturbing key properties. All of these transformations can be implemented as linear transformations, or affine (i.e. translated linear) transformations, and in particular, as nonsingular transformations. It is important to note that when such a transformation is applied to an ellipse, the resulting figure is again an ellipse.

Establishing that this is true is surprisingly involved. Although a complete proof will not be presented here, the following discussion should provide a sense of what the proof entails.

So, what is involved in this question? First, a translation simply moves a figure from one location to another, without changing any aspect of the size or shape. Therefore, translating an ellipse definitely produces an ellipse, and one that is congruent to the original. Accordingly, we can focus on linear (as opposed to affine) transformations.

Second, it is enough to show that linear transformations always take a circle to an ellipse. After all, we know that we can always take an ellipse to a circle by stretching or shrinking in one direction, and that is given by a linear transformation (see Representing transformations with matrices). We can incorporate this process into a chain of operations to represent any linear transformation. In greater detail: suppose we want to apply a matrix A to every point of some ellipse. And suppose that the matrix B takes that ellipse to a circle, with B−1 the matrix for the inverse transformation. Then apply (AB−1)B to the ellipse. This has the same effect as just applying A. On the other hand, it first changes the ellipse to a circle, and then applies the linear transformation AB−1to that. If every linear transformation takes a circle to an ellipse, the end result of applying A to the ellipse will be another ellipse. This idea is shown schematically in Figure 1 below.

Figure 1.  Combining Transformations.

Figure 1

Third, every linear transformation is continuous. Indeed, if (u, v) is given by applying a linear transformation to (x, y), then u and v are each linear functions of x and y and hence continuous. That implies that the overall transformation is continuous. Accordingly, applying a nonsingular linear transformation to an ellipse must produce a curve that is bounded, closed, connected, and which contains a two-dimensional interior. So we know immediately that a linear transformation takes an ellipse to some sort of oval.

Fourth, given the equation of an ellipse and the matrix for a linear transformation, it is possible to determine the equation for the transformed image of the ellipse, and both the original ellipse and its transformed image are given by quadratic equations in x and y (details here). These are equations of the form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, where A, B, C, D, E, and F represent fixed real constants. The standard equation of an ellipse x2/a2 + y2/b2 = 1 is an example of this type of equation, with A = 1/a2, C = 1/b2, F = −1, and all the other constants equal to zero. Nonzero values for D and E correspond to an ellipse not centered at the origin; a nonzero value of B corresponds to an ellipse whose axes are not aligned with the coordinate axes (details here). And performing a linear transformation on a curve given by a quadratic equation produces another curve also given by a quadratic equation. But not every quadratic equation represents an ellipse.

Now there are two ways to proceed. One approach involves analyzing all the curves given by quadratic equations. These turn out to be exactly the conic sections, which are ellipses, parabolas, hyperbolas, as well as some exceptional cases which can be a combination of lines or one or two individual points. But only ellipses take the appearance of an oval. Thus, by continuity, the transformed image of an ellipse must be an ellipse. Definitions and standard equations of the different conic sections are given in most calculus books. See [4, section 9.1] for one treatment, or visit Wolfram MathWorld. Verification that every quadratic equation reduces to one of the conic sections is outlined here.

An alternative approach is to analyze just the quadratic equations which correspond to ellipses. For this approach, it is shown that a quadratic equation Ax2 + Bxy + Cy2 + Dx + Ey = 1 represents an ellipse if and only if  A > 0, C > 0, and B2 − 4AC < 0. Then, it must also be shown that a nonsingular linear transformation of a curve with one such equation also produces a curve with another such equation. This can be done efficiently using matrix equations and properties. An overview is given here.