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 $a$ and $b$ of the resulting ellipse or ellipsoid is

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

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

Definitions

There are three variants: the flattening $f,$ ^[1] sometimes called the first flattening,^[2] as well as two other "flattenings" $f'$ and $n,$ 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, $a$ 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	$n$	${\frac {a-b}{a+b}}$	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}{\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 $e$ is the eccentricity.

References

^ Snyder, John P. (1987). Map Projections: A Working Manual. U.S. Geological Survey Professional Paper. Vol. 1395. Washington, D.C.: U.S. Government Printing Office. doi:10.3133/pp1395.
^ Tenzer, Róbert (2002). "Transformation of the Geodetic Horizontal Control to Another Reference Ellipsoid". Studia Geophysica et Geodaetica. 46 (1): 27–32. doi:10.1023/A:1019881431482. S2CID 117114346. ProQuest 750849329.
^ For example, $f'$ is called the second flattening in: Taff, Laurence G. (1980). An Astronomical Glossary (Technical report). MIT Lincoln Lab. p. 84.
However, $n$ 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.
^ Maling, Derek Hylton (1992). Coordinate Systems and Map Projections (2nd ed.). Oxford; New York: Pergamon Press. p. 65. ISBN 0-08-037233-3.
Rapp, Richard H. (1991). Geometric Geodesy, Part I (Technical report). Ohio State Univ. Dept. of Geodetic Science and Surveying.
Osborne, P. (2008). "The Mercator Projections" (PDF). §5.2. Archived from the original (PDF) on 2012-01-18.
^ Lapaine, Miljenko (2017). "Basics of Geodesy for Map Projections". In Lapaine, Miljenko; Usery, E. Lynn (eds.). Choosing a Map Projection. Lecture Notes in Geoinformation and Cartography. pp. 327–343. doi:10.1007/978-3-319-51835-0_13. ISBN 978-3-319-51834-3.
Karney, Charles F.F. (2023). "On auxiliary latitudes". Survey Review: 1–16. arXiv:2212.05818. doi:10.1080/00396265.2023.2217604. S2CID 254564050.
^ 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

[snyder-1] Snyder, John P. (1987). Map Projections: A Working Manual. U.S. Geological Survey Professional Paper. Vol. 1395. Washington, D.C.: U.S. Government Printing Office. doi:10.3133/pp1395.

[2] Tenzer, Róbert (2002). "Transformation of the Geodetic Horizontal Control to Another Reference Ellipsoid". Studia Geophysica et Geodaetica. 46 (1): 27–32. doi:10.1023/A:1019881431482. S2CID 117114346. ProQuest 750849329.

[3] For example, $f'$ is called the second flattening in: Taff, Laurence G. (1980). An Astronomical Glossary (Technical report). MIT Lincoln Lab. p. 84.
However, $n$ 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.

[4] Maling, Derek Hylton (1992). Coordinate Systems and Map Projections (2nd ed.). Oxford; New York: Pergamon Press. p. 65. ISBN 0-08-037233-3.
Rapp, Richard H. (1991). Geometric Geodesy, Part I (Technical report). Ohio State Univ. Dept. of Geodetic Science and Surveying.
Osborne, P. (2008). "The Mercator Projections" (PDF). §5.2. Archived from the original (PDF) on 2012-01-18.

[5] Lapaine, Miljenko (2017). "Basics of Geodesy for Map Projections". In Lapaine, Miljenko; Usery, E. Lynn (eds.). Choosing a Map Projection. Lecture Notes in Geoinformation and Cartography. pp. 327–343. doi:10.1007/978-3-319-51835-0_13. ISBN 978-3-319-51834-3.
Karney, Charles F.F. (2023). "On auxiliary latitudes". Survey Review: 1–16. arXiv:2212.05818. doi:10.1080/00396265.2023.2217604. S2CID 254564050.

[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