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Flattening

Measure of compression between circle to ellipse or sphere to an ellipsoid of revolution

a}} compressed to an ellipse.

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 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, sometimes called the first flattening, as well as two other "flattenings" f' and 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 is the larger dimension (e.g. semimajor axis), whereas b is the smaller (semiminor axis). All flattenings are zero for a circle (). ::{| class="wikitable" ! style="padding-left: 0.5em" scope="row" | (First) flattening |- ! style="padding-left: 0.5em" scope="row" | Second flattening |- ! style="padding-left: 0.5em" scope="row" | Third flattening |}

Identities

The flattenings can be related to each-other:

:\begin{align} f = \frac{2n}{1 + n}, \[5mu] n = \frac{f}{2 - f}. \end{align}

The flattenings are related to other parameters of the ellipse. For example, :\begin{align} \frac ba &= 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{align}

where e is the eccentricity.

References

Snyder, John P.. (1987). "Map Projections: A Working Manual". U.S. Government Printing Office.
Taff, Laurence G.. (1980). "An Astronomical Glossary". MIT Lincoln Lab.
Rapp, Derek Hylton. (1992). ["Geometric Geodesy, Part I"](http://hdl.handle.net/1811/24333}} {{pb}} {{cite web). Ohio State Univ. Dept. of Geodetic Science and Surveying.
Lapaine, Miljenko. (2017). "Choosing a Map Projection". Survey Review.
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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celestial-mechanics geodesy trigonometry circles ellipsoids

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