Is there a general formula for the eccentricity $e$ of ellipsoid?

Question

For an ellipse of eccentricity $e$ the formulas are:

${x^2 \over a^2} + {y^2 \over b^2} = 1 \\ e = \sqrt {1-\left({b \over a} \right)^2}$

what about the "3D case"?

Answer 1

There is no 3D analogue of eccentricity, in the sense that there isn't a single real parameter that uniquely characterizes the shape up to scale. An $n$-dimensional ellipsoid's shape can be characterized up to rotations and reflections by $n$ real parameters, the lengths of its principal axes. These generalize the semimajor and semiminor axes, and for an ellipsoid with equation

$$\sum_{i=1}^n \left( \frac{x_i}{\sigma_i} \right)^2 = 1$$

are given by the denominators $\sigma_i$. More generally, for an ellipsoid with equation $x^T M x = 0$ where $x = \left[ \begin{array}{c} x_1 \\ \vdots \\ x_n \end{array} \right]$ and $M$ is a positive-definite symmetric matrix, the principal axes are the lines spanned by the eigenvectors of $M$ and the lengths of the principal axes are the inverses of the corresponding eigenvalues.

An ellipsoid is characterized up to rotations, reflections, and scaling by the ratios $\frac{\sigma_i}{\sigma_j}$ among the lengths of its principal axes; it suffices to consider the $n-1$ ratios $\frac{\sigma_i}{\sigma_{i+1}}$, arranging the lengths in order $\sigma_1 \ge \sigma_2 \ge \dots \ge \sigma_n$. Only when $n = 2$ does this boil down to a single number.

Quadrics which are not ellipses also have principal axes although the corresponding eigenvalues may be negative or zero, so the corresponding "lengths" may be negative or infinite.

Answer 2

In the aspect of measure of the boundary

Perimeter of an ellipse

\begin{align} 1 &= \frac{x^2}{a^2}+\frac{y^2}{b^2} \tag{$a \ge b$} \\ P &= 4aE(e_{ab}) \\ e_{ab} &= \frac{\sqrt{a^2-b^2}}{a} \\ \end{align}

Surface area of an ellipsoid

\begin{align} 1 &= \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} \tag{$a \ge b \ge c$} \\ S &= 2\pi \left[ c^2+\frac{bc^2}{\sqrt{a^2-c^2}}F(\theta,k)+b\sqrt{a^2-c^2} E(\theta,k) \right] \\ \theta &= \cos^{-1} \frac{c}{a} \\ k &= \frac{a}{b} \sqrt{\frac{b^2-c^2}{a^2-c^2}} \\ &= \frac{e_{bc}}{e_{ac}} \\ k' &= \sqrt{1-k^2} \\ &= \frac{c}{b} \sqrt{\frac{a^2-b^2}{a^2-c^2}} \\ &= \frac{e_{ab}}{e_{ac}} \end{align}

• For prolate spheroid

$$b=c \implies (k,k')=(0,1)$$

• For oblate spheroid

$$a=b \implies (k,k')=(1,0)$$

• $k$ or $k'$ is not quite well-defined for the case of sphere.

In the aspect of confocal system

Confocal conics

\begin{align} 1 &= \frac{x^2}{a^2+s}+\frac{y^2}{b^2+s} \\ e^2 &= \frac{a^2-b^2}{a^2+s} \end{align}

Confocal quadrics

\begin{align} 1 &= \frac{x^2}{a^2+s}+\frac{y^2}{b^2+s}+\frac{z^2}{c^2+s} \\ \kappa^2 &= \frac{a^2-c^2}{a^2+s} \end{align}

• For $s\to \infty$, $\kappa \to 0$ which close to a sphere.

• For $s\to -c$, $\kappa \to 1$ which shrinks to focal ellipse namely

$$0=z=\frac{x^2}{a^2-c^2}+\frac{y^2}{b^2-c^2}-1$$

• This can not analogous to paraboloids.

See also another posts of mine here and here.