# 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).