# Flattening Research Materials

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Flattening Images

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Rihanna - Take A Bow
Music video by Rihanna performing Take A Bow. YouTube view counts pre-VEVO: 66288884. (C) 2008 The Island Def Jam Music Group.
Key & Peele: Substitute Teacher
A substitute teacher from the inner city refuses to be messed with while taking attendance.
David Guetta - Just One Last Time ft. Taped Rai
"Just One Last Time" feat. Taped Rai. Available to download on iTunes including remixes of : Tiësto, HARD ROCK SOFA & Deniz Koyu http://smarturl.it/DGJustOne...
Steve Jobs vs Bill Gates. Epic Rap Battles of History Season 2.
Download This Song: http://bit.ly/KzLBGB Click to Tweet this Vid-ee-oh! http://bit.ly/Nt9lg8 Hi. My name is Nice Peter, and this is EpicLLOYD, and this is th...
MACKLEMORE & RYAN LEWIS - CAN'T HOLD US FEAT. RAY DALTON (OFFICIAL MUSIC VIDEO)
Macklemore & Ryan Lewis present the official music video for Can't Hold Us feat. Ray Dalton. Can't Hold Us on iTunes: https://itunes.apple.com/us/album/cant-...
Draw My Life- Jenna Marbles
This video accidentally turned out kind of sad, ME SO SOWWY IT NOT POSED TO BE SAD WHO WANTS HUGS AND COOKIES? Also, FYI for anyone attempting this, it takes...
F*@#ing Ben Affleck
Jimmy reveals that he is f*@#ing Ben Affleck.
Fast Food Lasagna - Epic Meal Time
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Draw My Life - Ryan Higa
So i was pretty hesitant to make this video... but after all of your request, here is my Draw My Life video! Check out my 2nd Channel for more vlogs: http://...
Jack Sparrow (feat. Michael Bolton)
Buy at iTunes: http://goo.gl/zv4o9. New album on sale now! http://turtleneckandchain.com.
Giant 6ft Water Balloon - The Slow Mo Guys
Follow on Twitter! - https://twitter.com/#!/GavinFree Watch this one in HD! The slow mo guys are well aware that water balloons are always good in slow motio...
Master Chief vs Leonidas. Epic Rap Battles of History Season 2.
download this song: http://bit.ly/ERB17 click to tweet this vid-ee-oh! http://clicktotweet.com/vCJ_8 This. Is. Merchandise: http://bit.ly/ERBMerch Hi. My nam...
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

In the following, a is the larger dimension (e.g. semimajor axis), while b is the smaller (semiminor axis). All flattenings are zero for a circle (a=b).

 (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.[citation needed])

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 (2nd ed.). Oxford; New York: Pergamon Press. ISBN 0-08-037233-3.
2. ^ Snyder, John P. (1987). Map Projections: A Working Manual. U.S. Geological Survey Professional Paper 1395. Washington, D.C.: United States Government Printing Office.
3. ^ Torge, W. (2001). Geodesy (3rd edition). 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-print arXiv:0908.1824, Bibcode1825AN......4..241B
7. ^
8. ^ Isaac Newton:Principia Book III Proposition XIX Problem III, p. 407 in Andrew Motte translation, available on line at [2]
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