Definition of an ellipsoid based on its focal points

For confocal central quadrics,

$$\frac{x^2}{a^2+s}+\frac{y^2}{b^2+s}+\frac{z^2}{c^2+s}=1$$

where $a>b>c\geq 0$, there're one focal ellipse namely,

$$\left \{ \begin{array}{rcl} \displaystyle \frac{x^2}{a^2-c^2}+\frac{y^2}{b^2-c^2} &=& 1 \\ z &=& 0 \end{array} \right.$$

and one focal hyperbola namely,

$$\left \{ \begin{array}{rcl} \displaystyle \frac{x^2}{a^2-b^2}-\frac{z^2}{b^2-c^2} &=& 1 \\ y &=& 0 \end{array} \right.$$

For confocal paraboloids,

$$\frac{x^2}{a^2+s}+\frac{y^2}{b^2+s}=z+\frac{s}{4}$$

where $a>b\geq 0$, there're two focal parabolae namely,

$$\left \{ \begin{array}{rcl} \displaystyle \frac{x^2}{a^2-b^2} &=& \displaystyle z-\frac{b^2}{4} \\ y &=& 0 \end{array} \right.$$

and

$$\left \{ \begin{array}{rcl} \displaystyle \frac{y^2}{b^2-a^2} &=& \displaystyle z-\frac{a^2}{4} \\ x &=& 0 \end{array} \right.$$

There's so-called thread construction, please see p.19 of this onwards.

Note that the two focal conics are called conjugate conics and their eccentricities are reciprocal to each other.

Any point on the conjugate conics is the radiant point of the enveloping cone for that conic and of its Dandelin sphere.

P.S.:

The lines of curvature of an ellipsoid actually are the geodesic ellipses and geodesic hyperbolae from two adjacent umbilical points.

See more details here.