Flattening

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 of the resulting ellipse or ellipsoid is

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

The compression factor is ${\displaystyle {\frac {b}{a}}\,\!}$ 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.^[1][2][3] and online web texts^[4][5]

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	${\displaystyle f}$	${\displaystyle {\frac {a-b}{a}}}$	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	${\displaystyle n,\quad (f'')}$	${\displaystyle {\frac {a-b}{a+b}}}$	Used in geodetic calculations as a small expansion parameter.[6]

Identities[]

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

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

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

See also[]

• Astronomy
• Earth flattening
• Earth's rotation
• Eccentricity (mathematics) § Ellipses
• Equatorial bulge
• Gravitational field
• Gravity formula
• Ovality
• Planetology
• Planetary flattening
• Sphericity
• Roundness (object)
• Darwin–Radau equation

References[]