# Flattening

(Redirected from Oblateness)

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

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

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

## Definitions

There are three variants: the flattening ${\displaystyle f,}$ [1] sometimes called the first flattening,[2] as well as two other "flattenings" ${\displaystyle f'}$  and ${\displaystyle n,}$  each sometimes called the second flattening,[3] sometimes only given a symbol,[4] or sometimes called the second flattening and third flattening, respectively.[5]

In the following, ${\displaystyle a}$  is the larger dimension (e.g. semimajor axis), whereas ${\displaystyle b}$  is the smaller (semiminor axis). All flattenings are zero for a circle (a = b).

(First) flattening  Second flattening Third flattening ${\displaystyle f}$ ${\displaystyle {\frac {a-b}{a}}}$ Fundamental. Geodetic reference ellipsoids are specified by giving ${\displaystyle {\frac {1}{f}}\,\!}$ ${\displaystyle f'}$ ${\displaystyle {\frac {a-b}{b}}}$ Rarely used. ${\displaystyle n}$ ${\displaystyle {\frac {a-b}{a+b}}}$ Used in geodetic calculations as a small expansion parameter.[6]

## Identities

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

3. ^ For example, ${\displaystyle f'}$  is called the second flattening in: Taff, Laurence G. (1980). An Astronomical Glossary (Technical report). MIT Lincoln Lab. p. 84.
However, ${\displaystyle n}$  is called the second flattening in: Hooijberg, Maarten (1997). Practical Geodesy: Using Computers. Springer. p. 41. doi:10.1007/978-3-642-60584-0_3.