# 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 of the resulting ellipse or ellipsoid is

$\mathrm {flattening} =f={\frac {a-b}{a}}.$ The compression factor is ${\frac {b}{a}}\,\!$ in each case; for the ellipse, this is also its aspect ratio.

## Definitions

There are three variants of flattening; when it is necessary to avoid confusion, the main flattening is called the first flattening. and online web texts

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. $n,\quad (f'')\,\!$ ${\frac {a-b}{a+b}}\,\!$ Used in geodetic calculations as a small expansion parameter.

## Identities

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

{\begin{aligned}b&=a(1-f)=a\left({\frac {1-n}{1+n}}\right),\\e^{2}&=2f-f^{2}={\frac {4n}{(1+n)^{2}}}.\\\end{aligned}} where $e$ is the eccentricity.