Flattening

"Ellipticity" redirects here. For ellipticity in differential calculus, see elliptic operator. For other uses, see Flattening (disambiguation).

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 ${\displaystyle f}$ 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	${\displaystyle f}$	${\displaystyle {\frac {a-b}{a}}}$	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	${\displaystyle n}$	${\displaystyle {\frac {a-b}{a+b}}}$	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.

See also

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

References

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, ${\displaystyle f'}$ 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.
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.
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