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.

(This post first appeared at my other (now deleted) blog, and was transferred to this blog on 25 January 2021.)