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Measure of compression between circle to ellipse or sphere to an ellipsoid of revolution
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 and its definition in terms of the semi-axes of the resulting ellipse or ellipsoid is
The compression factor is b/a in each case. For the ellipse, this factor is also the aspect ratio of the ellipse.
There are two other variants of flattening (see below) and when it is necessary to avoid confusion the above flattening is called the first flattening. The following definitions may be found in standard texts and online web texts
Definitions of flattening
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).
For the WGS84 ellipsoid to model Earth, the defining values are
a (equatorial radius): 6 378 137.0 m
1/f (inverse flattening): 298.257 223 563
from which one derives
b (polar radius): 6 356 752.3142 m,
so that the difference of the major and minor semi-axes is 21.385 km (13 mi). (This is only 0.335% of the major axis, so a representation of Earth on a computer screen would be sized as 300px by 299px. Because this would be virtually indistinguishable from a sphere shown as 300px by 300px, illustrations typically greatly exaggerate the flattening in cases where the image needs to represent Earth's oblateness.)
Other values in the Solar System are Jupiter, f = 1/16; Saturn, f = 1/10, the Moonf = 1/900. The flattening of the Sun is about .
^Rapp, Richard H. (1991). Geometric Geodesy, Part I. Dept. of Geodetic Science and Surveying, Ohio State Univ., Columbus, Ohio. 
^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