Flattening

There are three variants of flattening; when it is necessary to avoid confusion, the main flattening is called the first flattening. The following definitions may be found in standard texts[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]

The flattenings can be related to each-other:

${\displaystyle {\begin{aligned}f={\frac {2n}{1+n}},\\[5mu]n={\frac {f}{2-f}}.\end{aligned}}}$

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

${\displaystyle {\begin{aligned}b&=a(1-f)=a\left({\frac {1-n}{1+n}}\right),\\[5mu]e^{2}&=2f-f^{2}={\frac {4n}{(1+n)^{2}}},\\[5mu]f&=1-{\sqrt {1-e^{2}}},\end{aligned}}}$

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

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