# Flattening

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The oblateness, ellipticity, or flattening of an oblate spheroid, or oblatum, is a measure of the "squashing" of the spheroid's Geographical pole, towards its equator. If $a$ is the distance from the spheroid center to the equator——the transverse radius——and $b$ the distance from the center to the pole——the conjugate radius——then $f\mbox{lattening}=\quad\frac{a-b}{a}\,\!$.

## First, second and third flatteningEdit

The first, primary flattening, f, is the versine of the spheroid's angular eccentricity, $o\!\varepsilon\,\!$:

$o\!\varepsilon=\arccos\left(\frac{b}{s}\right)=2\arctan\left(\!\sqrt{\frac{a-b}{a+b}}\;\right);\,\!$
$f=\mbox{ver}(o\!\varepsilon)=2\sin^2\left(\frac{o\!\varepsilon}{2}\right)=1-\cos(o\!\varepsilon)=\frac{a-b}{a};\,\!$
• The flattening ratio for Earth is 1:298.257223563[1] (which corresponds to a radius difference of 21.385 km of the Earth radius 6378.137 - 6356.752 km) and would not be realized visually from space, since the difference represents only 0.335 %.
• The flattening of Jupiter (1:16) and Saturn (nearly 1:10), in contrast, can be seen even in a small telescope;
• Conversely, that of the Sun is less than 1:1000 and that of the Moon barely 1:900.

The amount of flattening depends on

• the relation between gravity and centrifugal force;

and in detail on

• size and density of the celestial body;
• the rotation of the planet or star;
• and the elasticity of the body.

There is also a second flattening, f' ,

$f'=\frac{\mbox{ver}(o\!\varepsilon)}{1-\mbox{ver}(o\!\varepsilon)} =\frac{2\sin^2\left(\frac{o\!\varepsilon}{2}\right)}{1-2\sin^2\left(\frac{o\!\varepsilon}{2}\right)}=\sec(o\!\varepsilon)-1 =\frac{1-\cos(o\!\varepsilon)}{\cos(o\!\varepsilon)} =\frac{a-b}{b};\,\!$

and a third flattening[2][3], f'' (more commonly denoted as "n" and first used in 1837 by Friedrich Bessel on calculation of meridian arc length[4]), that is the squared half-angle tangent of $o\!\varepsilon\,\!$:

$f''=\frac{\mbox{ver}(o\!\varepsilon)}{\mbox{vrc}(o\!\varepsilon)} =\frac{\sin^2\left(\frac{o\!\varepsilon}{2}\right)}{1-\sin^2\left(\frac{o\!\varepsilon}{2}\right)} =\tan^2\left(\frac{o\!\varepsilon}{2}\right) =\frac{1-\cos(o\!\varepsilon)}{1+\cos(o\!\varepsilon)} =\frac{a-b}{a+b}.\,\!$

## Prolate valuationsEdit

The above formations apply to an ellipse and oblatum, which is an ellipse rotated about its polar, or conjugate, axis, resulting in $a>b\,\!$. If it is rotated about its equatorial, or transverse, axis, it is a prolate spheroid, or prolatum, where $b>a\,\!$.
With a prolatum, $a\,\!$ and $b\,\!$ are reversed in all of the flattening formation elements, except for the denominators of f and f' , which means their $o\!\varepsilon\,\!$ function assignments are reversed[5]:

$o\!\varepsilon=\arccos\left(\frac{a}{b}\right)=2\arctan\left(\!\sqrt{\frac{b-a}{b+a}}\;\right);\,\!$
$\begin{matrix}{}_{}\\\; f&=&\frac{b-a}{a} &=&\sec(o\!\varepsilon)-1 &=&\frac{1-\cos(o\!\varepsilon)}{\cos(o\!\varepsilon)} &=&\frac{2\sin^2\left(\frac{o\!\varepsilon}{2}\right)}{1-2\sin^2\left(\frac{o\!\varepsilon}{2}\right)};\\\\ f'&=&\frac{b-a}{b} &=&1-\cos(o\!\varepsilon) &=&\frac{1-\cos(o\!\varepsilon)}{1} &=&2\sin^2\left(\frac{o\!\varepsilon}{2}\right);\\\\ f''&=&\frac{b-a}{b+a} &=&\tan^2\left(\frac{o\!\varepsilon}{2}\right) &=&\frac{1-\cos(o\!\varepsilon)}{1+\cos(o\!\varepsilon)} &=&\frac{2\sin^2\left(\frac{o\!\varepsilon}{2}\right)}{2-2\sin^2\left(\frac{o\!\varepsilon}{2}\right)}.\\{}^{}\end{matrix}\,\!$

## NotesEdit

1. H. MORITZ (1979): Report of Special Study Group N° 539 of I.A.G., Fundamental Geodetic Constants, presented at XVII General Assembly og I.U.G.G., Canberra. [PDF]
2. König, R. and Weise, K. H. (1951): Mathematische Grundlagen der höheren Geodäsie und Kartographie, Band 1, Das Erdsphäroid und seine konformen Abbildungen, Springer-Verlag, Berlin/Göttingen/Heidelberg, Sec I.1
3. Ганьшин, В. Н. (1967): Геометрия земного эллипсоида, Издательство «Недра», Москва
4. Bessel, F. W. (1837): Bestimmung der Axen des elliptischen Rotationssphäroids, welches den vorhandenen Messungen von Meridianbögen der Erde am meisten entspricht, Astronomische Nachrichten, 14, 333-346. [PDF]
5. MathWorld: Flattening.

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