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Flattening

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Measure of compression between circle to ellipse or sphere to an ellipsoid of revolution "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 f {\displaystyle f} and its definition in terms of the semi-axes a {\displaystyle a} and b {\displaystyle b} of the resulting ellipse or ellipsoid is

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

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

Definitions

There are three variants: the flattening f , {\displaystyle f,} sometimes called the first flattening, as well as two other "flattenings" f ′ {\displaystyle f'} and n , {\displaystyle n,} each sometimes called the second flattening, sometimes only given a symbol, or sometimes called the second flattening and third flattening, respectively.

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

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

Identities

The flattenings can be related to each-other:

f = 2 n 1 + n , n = f 2 − f . {\displaystyle {\begin{aligned}f={\frac {2n}{1+n}},\\n={\frac {f}{2-f}}.\end{aligned}}}

The flattenings are related to other parameters of the ellipse. For example,

b a = 1 − f = 1 − n 1 + n , e 2 = 2 f − f 2 = 4 n ( 1 + n ) 2 , f = 1 − 1 − e 2 , {\displaystyle {\begin{aligned}{\frac {b}{a}}&=1-f={\frac {1-n}{1+n}},\\e^{2}&=2f-f^{2}={\frac {4n}{(1+n)^{2}}},\\f&=1-{\sqrt {1-e^{2}}},\end{aligned}}}

where e {\displaystyle e} is the eccentricity.

See also

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, f ′ {\displaystyle f'} is called the second flattening in: Taff, Laurence G. (1980). An Astronomical Glossary (Technical report). MIT Lincoln Lab. p. 84. However, n {\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
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