Dandelin spheres

In geometry, the Dandelin spheres are one or two spheres that are tangent both to a plane and to a cone that intersects the plane. The intersection of the cone and the plane is a conic section, and the point at which either sphere touches the plane is a focus of the conic section, so the Dandelin spheres are also sometimes called focal spheres.

The Dandelin spheres were discovered in 1822. They are named in honor of the Belgian mathematician Germinal Pierre Dandelin, though Adolphe Quetelet is sometimes given partial credit as well. The Dandelin spheres can be used to prove at least two important theorems. Both of those theorems were known for centuries before Dandelin, but he made it easier to prove them.

The first theorem is that a closed conic section (i.e. an ellipse) is the locus of points such that the sum of the distances to two fixed points (the foci) is constant. This was known to Ancient Greek mathematicians such as Apollonius of Perga, but the Dandelin spheres facilitate the proof.

The second theorem is that for any conic section, the distance from a fixed point (the focus) is proportional to the distance from a fixed line (the directrix), the constant of proportionality being called the eccentricity. Again, this theorem was known to the Ancient Greeks, such as Pappus of Alexandria, but the Dandelin spheres facilitate the proof.

A conic section has one Dandelin sphere for each focus. In particular, an ellipse has two Dandelin spheres, both touching the same of the cone. A hyperbola has two Dandelin spheres, touching opposite nappes of the cone. A parabola has just one Dandelin sphere.

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