Minor Axis of Ellipsoid is Axis of Symmetry

Theorem

Let $\EE$ be an ellipsoid.

The minor axis of $\EE$ is an axis of symmetry of $\EE$.

Let us arrange $\EE$ such that the minor axis is aligned with the $x$-axis, say.

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

where $2 a$ is the length of the minor axis.

Let $R$ denote the space rotation of $180 \degrees$ about the $x$-axis, that is, the minor axis of $\EE$.

Then:

$\ds $

$\ds \map R {\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} + \dfrac {z^2} {c^2} }$

$\ds $

$=$

$\ds \dfrac {x^2} {a^2} + \dfrac {\paren {-y}^2} {b^2} + \dfrac {\paren {-z}^2} {c^2}$

$\ds $

$=$

$\ds \dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} + \dfrac {z^2} {c^2}$

Hence $R$ is demonstrated to be a symmetry mapping, and the result follows.

$\blacksquare$