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 and its definition in terms of the semi-axes and of the resulting ellipse or ellipsoid is
The compression factor is in each case; for the ellipse, this is also its aspect ratio.
Definitions There are three variants: the flattening sometimes called the first flattening, as well as two other "flattenings" and each sometimes called the second flattening,For example, is called the second flattening in: However, is called the second flattening in: sometimes only given a symbol, or sometimes called the second flattening and third flattening, respectively.
In the following, is the larger dimension (e.g. semimajor axis), whereas 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" | | style="padding-left: 0.5em" | | style="padding-left: 0.5em " | Fundamental. Geodetic reference ellipsoids are specified by giving |- ! style="padding-left: 0.5em" scope="row" | Second flattening | style="padding-left: 0.5em" | | style="padding-left: 0.5em" | | style="padding-left: 0.5em" | Rarely used. |- ! style="padding-left: 0.5em" scope="row" | Third flattening | style="padding-left: 0.5em" | | style="padding-left: 0.5em" | | style="padding-left: 0.5em" | Used in geodetic calculations as a small expansion parameter.F. W. Bessel, 1825, Uber die Berechnung der geographischen Langen und Breiten aus geodatischen Vermessungen, Astron.Nachr., 4(86), 241–254, , 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 ,
Identities The flattenings can be related to each-other:
The flattenings are related to other parameters of the ellipse. For example,
where is the eccentricity.
See also Earth flattening
Equatorial bulge Ovality Planetary flattening Sphericity Roundness (object)
References