# User:Peter Mercator/Draft for flattening

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"Ellipticity" redirects here. For ellipticity in differential calculus, see elliptic operator.
This page is about geometry. For psychopathology, see flattening of affect.
A circle of radius a compressed to an ellipse.
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

$\mathrm{flattening} = f =\frac {a - b}{a}.$

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 [1] [2][3] and online web texts[4][5]

### Definitions of flattening

 (first) flattening $f\,\!$ $\frac{a-b}{a}\,\!$ Fundamental. The inverse 1/f is the normal choice for geodetic reference ellipsoids. second flattening $f'\,\!$ $\frac{a-b}{b}\,\!$ Rarely used. third flattening $n\quad(f'')\,\!$ $\frac{a-b}{a+b}\,\!$ Used in geodetic calculations as a small expansion parameter.[6]

### Identities involving flattening

The flattenings are related to other parameters of the ellipse. For example:

\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}

### Numerical values for planets

For the Earth modelled by the WGS84 ellipsoid the defining values are[7]

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.)

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 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).[8] The amount of flattening depends on the density and the balance of gravitational force and centrifugal force.

## References

1. ^ Maling, Derek Hylton (1992). Coordinate Systems and Map Projections (second ed.). Pergamon Press. ISBN 0-08-037033-3 Check |isbn= value (help)..
2. ^ 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.
3. ^ Torge, W (2001) Geodesy (3rd edition), published by de Gruyter, isbn=3-11-017072-8
4. ^ Osborne, P (2008). The Mercator Projections Chapter 5.
5. ^ Rapp, Richard H. (1991). Geometric Geodesy, Part I, Dept. of Geodetic Science and Surveying, Ohio State Univ., Columbus, Ohio. [1]
6. ^ 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.
7. ^
8. ^ Isaac Newton:Principia Book III Proposition XIX Problem III, p. 407 in Andrew Motte translation, available on line at [2]