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Dandelin spheres |
In geometry, a nondegenerate conic section formed by a plane intersecting a cone has one or two Dandelin spheres characterized thus:
This concept is named in honor of Germinal Pierre Dandelin.
Each conic section has one Dandelin sphere for each focus.
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The reason for interest in Dandelin spheres is this theorem:
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 P1 and P2. As P moves along the ellipse, P1 and P2 move along the two circles. The distance from Fi to P is the same as the distance from Pi 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(P1, P) + d(P2, P) also remains constant. This follows from the fact that P lies on the straight line from P1 to P2, and the distance from P1 to P2 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.
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.
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.
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