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 {\displaystyle f} and its definition in terms of the semi-axes a {\displaystyle a} and b {\displaystyle b} of the resulting ellipse or ellipsoid is

f = a − b a . {\displaystyle f={\frac {a-b}{a}}.}

The compression factor is b / a {\displaystyle b/a} in each case; for the ellipse, this is also its aspect ratio.

Definitions

There are three variants: the flattening f , {\displaystyle f,} sometimes called the first flattening, as well as two other "flattenings" f ′ {\displaystyle f'} and n , {\displaystyle 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 {\displaystyle a} is the larger dimension (e.g. semimajor axis), whereas b {\displaystyle b} is the smaller (semiminor axis). All flattenings are zero for a circle (a = b).

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

Identities

The flattenings can be related to each-other:

f = 2 n 1 + n , n = f 2 − f . {\displaystyle {\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,

b a = 1 − f = 1 − n 1 + n , e 2 = 2 f − f 2 = 4 n ( 1 + n ) 2 , f = 1 − 1 − e 2 , {\displaystyle {\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 {\displaystyle e} is the eccentricity.

See also