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 French mathematician Germinal Pierre Dandelin, though Adolphe Quetelet is sometimes given partial credit as well. The Dandelin spheres can be used to give elegant modern proofs of two classical theorems known to Apollonius of Perga. 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. 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. A conic section has one Dandelin sphere for each focus. An ellipse has two Dandelin spheres touching the same nappe of the cone, while hyperbola has two Dandelin spheres touching opposite nappes. A parabola has just one Dandelin sphere. Consider the illustration, depicting a cone with apex S at the top. A plane e intersects the cone in a curve C (with blue interior). The following proof shall show that the curve C is an ellipse. The two brown Dandelin spheres, G1 and G2, are placed tangent to both the plane and the cone: G1 above the plane, G2 below. Each sphere touches the cone along a circle (colored white), and . Denote the point of tangency of the plane with G1 by F1, and similarly for G2 and F2 . Let P be a typical point on the curve C. To Prove: The sum of distances remains constant as the point P moves along the intersection curve C. (This is one definition of C being an ellipse, with and being its foci.) A line passing through P and the vertex S of the cone intersects the two circles, touching G1 and G2 respectively at points P1 and P2.