User:Peter Mercator/Draft for flattening
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
Identities involving flattening
The flattenings are related to other parameters of the ellipse. For example:
Numerical values for planets
- 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 the Earth on a computer screen could be sized as 300px by 299px. Since this would be indistinguishable from a sphere shown as 300px by 300px illustrations invariably greatly exaggerate the flattening.)
Origin of flattening
In 1687 Isaac Newton published the Principia in which he included a proof that a rotating self-gravitating fluid body in equilibrium takes the form of an oblate ellipsoid of revolution (a spheroid). The amount of flattening depends on the density and the balance of gravitational force and centrifugal force.
- Earth ellipsoid, Gravity flattening
- Equatorial bulge, Earth rotation
- Astronomy, Planetology
- Gravitational field, Gravity formula
- Maling, Derek Hylton (1992). Coordinate Systems and Map Projections (second ed.). Pergamon Press. ISBN 0-08-037033-3 Check
- Snyder, John P. (1987). Map Projections – A Working Manual. U.S. Geological Survey Professional Paper 1395. United States Government Printing Office, Washington, D.C.This paper can be downloaded from USGS pages.
- Torge, W (2001) Geodesy (3rd edition), published by de Gruyter, isbn=3-11-017072-8
- Osborne, P (2008). The Mercator Projections Chapter 5.
- 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-printarXiv:0908.1824, http://adsabs.harvard.edu/abs/1825AN......4..241B.
- WGS84 parameters are listed in the National Geospatial-Intelligence Agency publication TR8350.2 page 3-1.
- Isaac Newton:Principia Book III Proposition XIX Problem III, p. 407 in Andrew Motte translation, available on line at