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

Standard equation for an ellipse

Let us derive an equation for an ellipse obtained by stretching a circle, as discussed on the previous page. If the circle is centered at the origin with radius b, then the equation is x2 + y2 = b2. We will stretch by a factor of 2 horizontally, so that each point (h, k) is moved to (2h, k). Looked at differently, a transformed point (x, y) came from a starting position of (x/2, y). Then (x, y) is on the transformed image of the circle if and only if (x/2, y) satisfies the equation of the circle. That gives x2/4 + y2 = b2, or equivalently,

Had we stretched by a factor of r rather than 2, the equation would instead have been

This is the generic equation of an ellipse that has been stretched horizontally from a circle centered at the origin. It is customary to introduce the constant a = rb, so that the equation becomes

(1) .

This is the standard equation for an ellipse. The ellipse has x and y intercepts at ±a and ±b, respectively. The segments from a to a on the x axis and from b to b on the y axis are called the axes of the ellipse.

We derived the standard equation using a circle centered at the origin, and with a stretch parallel to the x axis. Of course, it is possible to draw an ellipse centered at any point of the plane, not just at the origin, and rotated to any angle so that its axes are not aligned with the coordinate axes. An equation for such an ellipse can be obtained from the standard form above by making a change of coordinates. On the other hand, given any ellipse in the plane, it is possible to impose a coordinate system with respect to which the ellipse is in the standard centered position. Relative to this coordinate system, the equation of the ellipse will take the standard form. This fact will be used to show on the next page that the definition of an ellipse involving two foci is equivalent to the definition based on stretching a circle.

Special Cases

If a = b the standard equation reduces to the equation of a circle, with radius given by the common value of a and b. We can still consider this to be an ellipse, although of a very special kind. It corresponds to stretching a circle by a factor of r = 1, which is to say, by no amount at all. The convention of considering a circle to also be an ellipse is a matter of convenience. It permits us to say, for example, that the standard equation defines an ellipse for any positive values of a and b, without having to include the added restriction that ab. Similarly, only by including circles among ellipses is it correct to state that applying an invertible linear transformation to an ellipse always results in another ellipse.

Another special case occurs if r = 0. This results in a degenerate ellipse. Note that the derivation above is not valid for r = 0, and a degenerate ellipse cannot be described by an instance of the standard equation.