thumb|right|200px |A circle of radius compressed to an ellipse.
thumb|right|200px |A sphere of radius 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 <math>f</math> and its definition in terms of the semi-axes <math>a</math> and <math>b</math> of the resulting ellipse or ellipsoid is
:<math> f =\frac {a - b}{a}.</math>
The compression factor is <math>b/a</math> in each case; for the ellipse, this is also its aspect ratio.
Definitions
There are three variants: the flattening <math>f,</math> sometimes called the first flattening, as well as two other "flattenings" <math>f'</math> and <math>n,</math> each sometimes called the second flattening, sometimes only given a symbol, or sometimes called the second flattening and third flattening, respectively.
In the following, <math>a</math> is the larger dimension (e.g. semimajor axis), whereas <math>b</math> is the smaller (semiminor axis). All flattenings are zero for a circle ().
::{| class="wikitable"
! style="padding-left: 0.5em" scope="row" | (First) flattening
| style="padding-left: 0.5em" | <math>f</math>
| style="padding-left: 0.5em" | <math>\frac{a-b}{a}</math>
| style="padding-left: 0.5em " | Fundamental. Geodetic reference ellipsoids are specified by giving <math>\frac{1}{f}\,\!</math>
|-
! style="padding-left: 0.5em" scope="row" | Second flattening
| style="padding-left: 0.5em" | <math>f'</math>
| style="padding-left: 0.5em" | <math>\frac{a-b}{b}</math>
| style="padding-left: 0.5em" | Rarely used.
|-
! style="padding-left: 0.5em" scope="row" | Third flattening
| style="padding-left: 0.5em" | <math>n</math>
| style="padding-left: 0.5em" | <math>\frac{a-b}{a+b}</math>
| style="padding-left: 0.5em" | Used in geodetic calculations as a small expansion parameter.
|}
Identities
The flattenings can be related to each-other:
:<math>\begin{align}
f = \frac{2n}{1 + n}, \\[5mu]
n = \frac{f}{2 - f}.
\end{align}</math>
The flattenings are related to other parameters of the ellipse. For example,
:<math>\begin{align}
\frac ba &= 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{align}</math>
where <math>e</math> is the eccentricity.
See also
- Earth flattening
- Equatorial bulge
- Ovality
- Planetary flattening
- Sphericity
- Roundness (object)