Can we Show the [Dandelin spheres ][1] scenario (shown by Hrhm for case of ellipse) for a hyperbola as well? Not happy with what we get to see of this on the net. 
 
The double nappe/ sheet of a cone is cut by a plane inclined at angle $ \beta < 2 \alpha $ ( cutting plane inclination to a generator of cone, cone vertex angle ) cutting both nappes with hyperbolas as intersecting arcs.

The Dandelin spheres are placed in each cone tangentially, cutting plane contacts spheres at respective foci so that their *difference* is constant.

I can also do this using ContourPlot3D, however you can make it with a great image quality.

Proper choice of $\alpha, \beta$ so that the difference of line segments can be convincingly visualized.

  [1]: http://math.stackexchange.com/questions/1833973/prove-that-the-directrix-focus-and-focus-focus-definitions-are-equivalent