Nowadays conic sections are not part of the standard high-school mathematics curriculum in Ontario (at least ellipses and hyperbolas are not; of course circles and parabolas are present), but they are interesting and important curves in mathematics, science, and engineering applications.

There are two ways to define an ellipse: (1) as the curve of intersection of a plane and a cone (for certain relative orientations), and (2) as a curve such that for each point *A* on the curve, the sum of the distances *A**F*_{1} and *A**F*_{2} is constant, where *F*_{1} and *F*_{2} are fixed points. Each point *F*_{1} and *F*_{2} is called a focus of the ellipse; it’s traditional to use Latin plural and call the two points together *foci* of the ellipse.

There is a beautiful connection between the two definitions of an ellipse via Dandelin spheres, named after Germinal Dandelin. (At least the spheres facilitate the proof that the two definitions are equivalent; the fact and its proof were known by the ancient Greeks.)

The basic geometry is this: Take a cone and place a sphere in it, so that the sphere is tangent to the cone along a circle *C*. Now introduce a plane that intersects the cone and is also tangent to the sphere at a point *F*_{1} not touching the circle *C*. The intersection of the plane with the cone is an ellipse, and sure enough, the point *F*_{1} is indeed a focus of the ellipse! Is this not delightful?

Now take the side of the plane opposite to the side where the sphere sits. It is possible to place a second sphere on this side of the plane that is both tangent to the plane and also tangent to (i.e. fits inside) the cone; the point of tangency of the second sphere and the plane is the second focus of the ellipse, the point *F*_{2}!

For a delightful (and brief) proof of the equivalence of the two definitions of the ellipse (which as a bonus contains a much better diagram than the ones in the Wikipedia links above), see here.

### Like this:

Like Loading...

*Related*

## About Santo D'Agostino

I have taught mathematics and physics since the mid 1980s. I have also been a textbook writer/editor since then. Currently I am working independently on a number of writing and education projects while teaching physics at my local university.
I love math and physics, and love teaching and writing about them. My blog also discusses education, science, environment, etc. https://qedinsight.wordpress.com
Further resources, and online tutoring, can be found at my other site
http://www.qedinfinity.com