# Flattening

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

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

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

## Identities

The flattenings can be related to each-other:

{\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,

{\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 $e$ is the eccentricity.