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2013-11-30T16:38:47Z

77.126.69.125: /* Mathematical description */

{{Redirect|Ellipticity|ellipticity in differential calculus|elliptic operator}}
{{About|geometry|psychopathology|flattening of affect}}
[[File:An ellipse with auxiliary circle.svg|thumb |right|200px |A circle of radius ''a'' compressed to an ellipse.]]
[[File:Ellipsoid revolution oblate aab auxiliary sphere.svg|thumb|right|200px |A sphere of radius ''a'' compressed to an oblate ellipsoid of revolution oblate.]]

'''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
::<math> \mathrm{flattening} = f =\frac {a - b}{a}.</math>

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 <ref name=maling>{{cite book | last=Maling |first=Derek Hylton | title=Coordinate Systems and Map Projections |edition=2nd |year=1992 | publisher =Pergamon Press|location=Oxford; New York |isbn=0-08-037233-3}}</ref><ref name=snyder>{{cite book |author=Snyder, John P. | title=Map Projections: A Working Manual |series=U.S. Geological Survey Professional Paper |volume=1395 | year=1987| publisher =United States Government Printing Office |location=Washington, D.C. |url=http://pubs.er.usgs.gov/pubs/pp/pp1395}}</ref><ref name=torge>Torge, W. (2001). ''Geodesy'' (3rd edition). de Gruyter. ISBN 3-11-017072-8</ref> and online web texts<ref name=osborne>Osborne, P. (2008). ''[http://mercator.myzen.co.uk/mercator.pdf The Mercator Projections]'' Chapter 5.</ref><ref name=rapp>Rapp, Richard H. (1991). ''Geometric Geodesy, Part I''. Dept. of Geodetic Science and Surveying, Ohio State Univ., Columbus, Ohio. [http://hdl.handle.net/1811/24333]</ref>

== 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'').
::{| class="wikitable" style="border: 1px solid darkgray; width: 60%;" cellpadding="5"
| style="padding-left: 0.5em"| (first) flattening
| style="padding-left: 0.5em"|<math>f\,\!</math>
| style="padding-left: 0.5em"|<math>\frac{a-b}{a}\,\!</math>
| style="width: 50%; padding-left: 0.5em "|Fundamental. The inverse 1/f is the normal choice for geodetic [[reference ellipsoid]]s.
|-
| style="padding-left: 0.5em"|second flattening
| style="padding-left: 0.5em"|<math>f'\,\!</math>
| style="padding-left: 0.5em"|<math>\frac{a-b}{b}\,\!</math>  
| style="padding-left: 0.5em"| Rarely used.
|-
| style="padding-left: 0.5em"| third flattening
| style="padding-left: 0.5em"|<math>n\quad(f'')\,\!</math>  
| style="padding-left: 0.5em"|<math>\frac{a-b}{a+b}\,\!</math>
| style="padding-left: 0.5em"| Used in geodetic calculations as a small expansion parameter.<ref name=bessel>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}}</ref>
|}

==Identities involving flattening==
The flattenings are related to other parameters of the ellipse. For example:
:<math>
\begin{align}
b&=a(1-f)=a\left(\frac{1-n}{1+n}\right),\\
e^2&=2f-f^2 = \frac{4n}{(1+n)^2}.\\
\end{align}
</math>

==Numerical values for planets==
For the [[Earth]] modelled by the [[WGS84]] ellipsoid the ''defining'' values are<ref>[http://earth-info.nga.mil/GandG/publications/tr8350.2/tr8350_2.htmlNIMAThe WGS84 parameters are listed in the National Geospatial-Intelligence Agency publication TR8350.2] page 3-1.</ref>
::''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 {{convert|21.385|km|0|abbr=on}}. (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. 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 the oblateness of the Earth.)

Other values in the Solar System are [[Jupiter]],  ''f''=1/16; [[Saturn]],  ''f''= 1/10, the [[Moon]]  ''f''= 1/900. The flattening of the [[Sun]] is less than 1/1000.

==Origin of flattening==
In 1687 [[Isaac Newton]] published the ''[[Philosophiæ Naturalis Principia Mathematica|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]]).<ref name=newton>Isaac Newton:''Principia'' Book III Proposition XIX Problem III, p. 407 in [https://archive.org/stream/ost-physics-newtonspmathema00newtrich/newtonspmathema00newtrich#page/n411/mode/2up Andrew Motte translation]</ref> The amount of flattening depends on the [[density]] and the balance of [[gravitational force]] and [[centrifugal force (rotating reference frame)|centrifugal force]].

== See also ==
* [[Astronomy]]
* [[Earth ellipsoid]]
* [[Earth's rotation]]
* [[Eccentricity (mathematics)#Ellipses|Eccentricity (mathematics)]]
* [[Equatorial bulge]]
* [[Gravitational field]]
* [[Gravity formula]]
* [[Ovality]]
* [[Planetology]]

==References==
{{Reflist}}

[[Category:Celestial mechanics]]
[[Category:Geodesy]]
[[Category:Trigonometry]]

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