Flattening
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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 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. ${\displaystyle n,\quad (f'')\,\!}$ ${\displaystyle {\frac {a-b}{a+b}}\,\!}$ 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.

## References

1. ^ Maling, Derek Hylton (1992). Coordinate Systems and Map Projections (2nd ed.). Oxford; New York: Pergamon Press. ISBN 0-08-037233-3.
2. ^ Snyder, John P. (1987). Map Projections: A Working Manual. U.S. Geological Survey Professional Paper. 1395. Washington, D.C.: United States Government Printing Office.
3. ^ Torge, W. (2001). Geodesy (3rd edition). de Gruyter. ISBN 3-11-017072-8
4. ^ Osborne, P. (2008). The Mercator Projections Archived 2012-01-18 at the Wayback Machine Chapter 5.
5. ^ Rapp, Richard H. (1991). Geometric Geodesy, Part I. Dept. of Geodetic Science and Surveying, Ohio State Univ., Columbus, Ohio. [1]
6. ^ F. W. Bessel, 1825, Uber die Berechnung der geographischen Langen und Breiten aus geodatischen Vermessungen, Astron.Nachr., 4(86), 241-254, doi:10.1002/asna.201011352, translated into English by C. F. F. Karney and R. E. Deakin as The calculation of longitude and latitude from geodesic measurements, Astron. Nachr. 331(8), 852-861 (2010), E-print arXiv:0908.1824, Bibcode:1825AN......4..241B