ellipticity

Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening is $f$ and its definition in terms of the semi-axes $a$ and $b$ of the resulting ellipse or ellipsoid is
:$f =\frac .$
The ''compression factor'' is $b/a$ in each case; for the ellipse, this is also its aspect ratio.

Definitions

There are three variants: the flattening $f,$ sometimes called the ''first flattening'', as well as two other "flattenings" $f'$ and $n,$ each sometimes called the ''second flattening'', sometimes only given a symbol, or sometimes called the ''second flattening'' and ''third flattening'', respectively.

In the following, $a$ is the larger dimension (e.g. semimajor axis), whereas $b$ is the smaller (semiminor axis). All flattenings are zero for a circle ().
::

Identities

The flattenings can be related to each-other:

:\begin
f = \frac, \\ mun = \frac.
\end

The flattenings are related to other parameters of the ellipse. For example,
:\begin
\frac ba &= 1-f = \frac, \\ mue^2 &= 2f-f^2 = \frac, \\ muf &= 1-\sqrt,
\end

where $e$ is the eccentricity.

See also

* Earth flattening
*
* Equatorial bulge
* Ovality
* Planetary flattening
* Sphericity
* Roundness (object)

References

{{reflist

Celestial mechanics
Geodesy
Trigonometry
Circles
Ellipsoids