Definition:Dandelin Spheres

Definition
Let $\CC$ be a double napped right circular cone with apex $O$.

Let $\PP$ be a plane which intersects $\CC$ such that:
 * $\PP$ does not pass through $O$
 * $\PP$ is not parallel to a generatrix of $\CC$
 * $\PP$ is not perpendicular to the axis of $\CC$.

Hence, by construction, the resulting conic section $\EE$ is either an ellipse or a hyperbola, and is not degenerate.

Let two spheres $\SS$ and $\SS'$ be constructed so that they have ring-contact with $\CC$ such that $\PP$ is tangent to both $\SS$ and $\SS'$.

Then $\SS$ and $\SS'$ are known as Dandelin spheres.

Also see

 * Dandelin's Theorem: $\PP$ touches $\SS$ and $\SS'$ at the foci of $\EE$.