The expressions for the geometrical cross-sections of rotating spheroids
can be obtained if we average the cross-sections 
(Eqs. (7 (click here)) - (8 (click here))) according to the equations similar to
Eqs. (13 (click here)), (15 (click here)).
For the 3D-orientation, we obtain (e=a/b)
for prolate spheroids and
for oblate spheroids.
For the perfect rotational orientation, the cross-sections are
for prolate spheroids and
for oblate ones. Here, E(m) is the complete elliptic integral of
the second kind.