Flattening

Short description: Measure of compression between circle to ellipse or sphere to an ellipsoid of revolution

A circle of radius a compressed to an ellipse.

A sphere of radius a compressed to an oblate ellipsoid of revolution.

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" $f^{\prime }$ 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	$f$	$\frac{a-b}{a}$	Fundamental. Geodetic reference ellipsoids are specified by giving $\frac{1}{f}$
Second flattening	$f^{\prime }$	$\frac{a-b}{b}$	Rarely used.
Third flattening	$n$	$\frac{a-b}{a+b}$	Used in geodetic calculations as a small expansion parameter.

Identities

The flattenings can be related to each-other:

$\begin{array}{r}f=\frac{2n}{1+n},\\ n=\frac{f}{2-f}.\end{array}$

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

$\begin{array}{rl}\frac{b}{a}& =1-f=\frac{1-n}{1+n},\\ e^2& =2f-f^2=\frac{4n}{(1+n{)}^2},\\ f& =1-\sqrt{1-e^2},\end{array}$

where $e$ is the eccentricity.

See also

• Earth flattening
• Eccentricity (mathematics) § Ellipses
• Equatorial bulge
• Ovality
• Planetary flattening
• Sphericity
• Roundness (object)