Flattening

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

\mathrm {flattening} =f={\frac {a-b}{a}}.

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

Identities

The flattenings can be related to each-other:

{\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,

{\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 $e$ is the eccentricity.

References

^ Maling, Derek Hylton (1992). Coordinate Systems and Map Projections (2nd ed.). Oxford; New York: Pergamon Press. ISBN 0-08-037233-3.
^ Snyder, John P. (1987). Map Projections: A Working Manual. U.S. Geological Survey Professional Paper. Vol. 1395. Washington, D.C.: United States Government Printing Office.
^ Torge, W. (2001). Geodesy (3rd edition). de Gruyter. ISBN 3-11-017072-8
^ Osborne, P. (2008). The Mercator Projections Archived 2012-01-18 at the Wayback Machine Chapter 5.
^ Rapp, Richard H. (1991). Geometric Geodesy, Part I. Dept. of Geodetic Science and Surveying, Ohio State Univ., Columbus, Ohio. [1]
^ 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

[maling-1] Maling, Derek Hylton (1992). Coordinate Systems and Map Projections (2nd ed.). Oxford; New York: Pergamon Press. ISBN 0-08-037233-3.

[snyder-2] Snyder, John P. (1987). Map Projections: A Working Manual. U.S. Geological Survey Professional Paper. Vol. 1395. Washington, D.C.: United States Government Printing Office.

[torge-3] Torge, W. (2001). Geodesy (3rd edition). de Gruyter. ISBN 3-11-017072-8

[osborne-4] Osborne, P. (2008). The Mercator Projections Archived 2012-01-18 at the Wayback Machine Chapter 5.

[rapp-5] Rapp, Richard H. (1991). Geometric Geodesy, Part I. Dept. of Geodetic Science and Surveying, Ohio State Univ., Columbus, Ohio. [1]

[bessel-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

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Definitions

Identities

See also

References