## Dandelin spheres

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 AF1 and AF2 is constant, where F1 and F2 are fixed points. Each point F1 and F2 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 F1 not touching the circle C. The intersection of the plane with the cone is an ellipse, and sure enough, the point F1 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 F2!

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.