Dandelin spheres

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Dandelin spheres—graphics by Hop David, used by permission

In geometry, a nondegenerate conic section formed by a plane intersecting a cone has one or two Dandelin spheres characterized thus:

Each Dandelin sphere touches, but does not cross, both the plane and the cone.

This concept is named in honor of Germinal Pierre Dandelin.

Each conic section has one Dandelin sphere for each focus.

An ellipse has two Dandelin spheres, both touching the same nappe of the cone.
A hyperbola has two Dandelin spheres, touching opposite nappes of the cone.
A parabola has just one Dandelin sphere.

1 Dandelin's theorem
- 1.1 Consequences of this theorem and its proof
- 1.2 The directrix in Dandelin's construction
2 External links

[edit] Dandelin's theorem

The reason for interest in Dandelin spheres is this theorem:

The point at which the sphere touches the plane is a focus of the conic section.

Proof: Consider the illustration, depicting a plane intersecting a cone in an ellipse. The two Dandelin spheres are shown. The intersection of each sphere with the cone is a circle. Each sphere touches the plane at a point. Call those two points F₁ and F₂. Let P be a typical point on the ellipse. The sum of distances d(F₁, P) + d(F₂, P) must be shown to remain constant as the point P moves along the curve. A line passing through P and the vertex of the cone intersects the two circles in points G₁ and G₂. As P moves along the ellipse, G₁ and G₂ move along the two circles. The distance from F_i to P is the same as the distance from G_i to P, because both are tangent to the same sphere. Consequently, the sum of distances d(F₁, P) + d(F₂, P) must be constant as P moves along the curve because the sum of distances d(G₁, P) + d(G₂, P) also remains constant. This follows from the fact that P lies on the straight line from G₁ to G₂, and the distance from G₁ to G₂ remains constant.

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.

[edit] Consequences of this theorem and its proof

If (as is often done) one takes the definition of the ellipse to be the locus of points P such that d(F₁, P) + d(F₂, P) = a constant, then the argument above proves that the intersection of a plane with a cone is indeed an ellipse. 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.

[edit] The directrix in Dandelin's construction

The directrix of a conic section can also be found using Dandelin's construction. Each Dandelin sphere intersects the cone at a circle; let both of these circles define their own planes. The intersections of these two planes with the conic section's plane will be two parallel lines; these lines form the directrices of the conic section. A parabola has only one Dandelin sphere, and thus has only one directrix.