Definition of an ellipsoid based on its focal points

I have a question concerning the formulation of an (3D) ellipsoid. The most common definition for an ellipsoid seems to be:

$E = \{ x=\left( x_1, \dots x_n \right)^T \in R^n: \sum_{i=1}^n \left( \frac{x_i}{r_i} \right)^2 = 1 \}$

based on the different radii $r_i$. However for an ellipse (2D) I also found the following definition:

$E = \{ x \in R^2: \| x - f_1 \| + \| x - f_2 \| = c \}$

which is based on the two focal points $f_1$ and $f_2$ of an ellipse. Now my question is, if there exist such kind of definitions also for ellipsoid in 3D or higher? How many focal points can an ellipsoid in 3D or higher actually have? I am especially interested in ellipsoids where all radii $r_1, \dots, r_n$ can be different (like a Scalene or triaxial ellipsoids).

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We should investigate if any biaxial (that is, by a plane containing any two ellipsoid axes) section of an ellipsoid gives an ellipse. Both answers give strange results, like having ellipse with four foci or with no foci at all. –  mbaitoff Feb 1 '11 at 11:17
If I remember correctly, the analogue of the pair of focal points for an ellipsoid in 3D are a pair of curves, namely an ellipse and a hyperbola (in two orthogonal planes). Unless someone else gives an answer, I will try to look into this and return later. –  Hans Lundmark Feb 1 '11 at 11:20
Hmm, maybe this is it: books.google.com/… –  Hans Lundmark Feb 1 '11 at 11:22