Dandelin spheres

[1][2] They are named in honor of the French mathematician Germinal Pierre Dandelin, though Adolphe Quetelet is sometimes given partial credit as well.

[3][4][5] The Dandelin spheres can be used to give elegant modern proofs of two classical theorems known to Apollonius.

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 plane e intersects the cone in a curve C (with blue interior).

remains constant as the point P moves along the intersection curve C. (This is one definition of C being an ellipse, with

That the intersection of the plane with the cone is symmetric about the perpendicular bisector of the line through F1 and F2 may be counterintuitive, but this argument makes it clear.

Adaptations of this argument work for hyperbolas and parabolas as intersections of a plane with a cone.

Another adaptation works for an ellipse realized as the intersection of a plane with a right circular cylinder.

[7] Ancient Greek mathematicians such as Pappus of Alexandria were aware of this property, but the Dandelin spheres facilitate the proof.

The first to do so may have been Pierce Morton in 1829,[8] or perhaps Hugh Hamilton who remarked (in 1758) that a sphere touches the cone at a circle which defines a plane whose intersection with the plane of the conic section is a directrix.

[1][9][10][11] The focus-directrix property can be used to prove that astronomical objects move along conic sections[broken anchor] around the Sun.

Two Dandelin spheres touch the pale yellow plane that intersects the cone. The points of tangency F 1 , F 2 are the foci of the blue ellipse. The spheres are also tangent to the cone at circles k 1 , k 2 . For a point P on the ellipse, the tangent segments PF 1 and PF 2 can each be reflected to other tangents of equal length, PF 1 = PP 1 and PF 2 = PP 2 , with PP 1 and PP 2 colinear along the ray SP. Their combined length P 1 P + PP 2 = P 1 P 2 = L is the distance between circles k 1 and k 2 , and is independent of the choice of P; thus any point on the ellipse has PF 1 + PF 2 = L.
This construction shows how the focal points of an ellipse can be found using the Dandelin spheres. The angle bisector between the line representing the plane and a line representing the cone surface leads to the center of the respective sphere.
Cylinder case