Dandelin Spheres for the Hyperbola

The hyperbola can be defined as the curve formed by the intersection of a plane with the two nappes of a cone. In this Demonstration, Dandelin's spheres show the relationship between a hyperbola and its foci and directrices.

Dandelin's spheres have a special relationship with their associated conic section: their tangency points with the plane cutting the cone correspond to the foci of the conic section, while the planes through their circles of contact with the cone intersect the cutting plane in the conic section's directrices.

To get a very responsive Demonstration, the hyperbola and the contact circles are represented as nonuniform rational B-splines, avoiding the use of Wolfram Language plotting functions.