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 ${\displaystyle f}$ and its definition in terms of the semi-axes ${\displaystyle a}$ and ${\displaystyle b}$ of the resulting ellipse or ellipsoid is

${\displaystyle f={\frac {a-b}{a)).}$

The compression factor is ${\displaystyle b/a}$ in each case; for the ellipse, this is also its aspect ratio.

Definitions

There are three variants: the flattening ${\displaystyle f,}$^[1] sometimes called the first flattening,^[2] as well as two other "flattenings" ${\displaystyle f'}$ and ${\displaystyle n,}$ each sometimes called the second flattening,^[3] sometimes only given a symbol,^[4] or sometimes called the second flattening and third flattening, respectively.^[5]

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

(First) flattening	${\displaystyle f}$	${\displaystyle {\frac {a-b}{a))}$	Fundamental. Geodetic reference ellipsoids are specified by giving ${\displaystyle {\frac {1}{f))\,\!}$
Second flattening	${\displaystyle f'}$	${\displaystyle {\frac {a-b}{b))}$	Rarely used.
Third flattening	${\displaystyle n}$	${\displaystyle {\frac {a-b}{a+b))}$	Used in geodetic calculations as a small expansion parameter.[6]

Identities

The flattenings can be related to each-other:

${\displaystyle {\begin{aligned}f={\frac {2n}{1+n)),\\[5mu]n={\frac {f}{2-f)).\end{aligned))}$

The flattenings are related to other parameters of the ellipse. For example,

${\displaystyle {\begin{aligned}{\frac {b}{a))&=1-f={\frac {1-n}{1+n)),\\[5mu]e^{2}&=2f-f^{2}={\frac {4n}{(1+n)^{2))},\\[5mu]f&=1-{\sqrt {1-e^{2))},\end{aligned))}$

where ${\displaystyle e}$ is the eccentricity.