# Flattening

A sphere of radius a compressed to an oblatum (top), and the polar radius b “stretched” to create an prolatum (bottom).

The flattening, ellipticity, or oblateness of an oblate spheroid, or oblatum, is a measure of the "squashing" of the spheroid's pole, towards its equator. If $a\,\!$ is the distance from the spheroid center to the equator and $b$ the distance from the center to the pole, then $a>b\,\!$; if $a then the meridians are the object of flattening and the polar radius is “stretched”, resulting in a prolate spheroid, or prolatum:

$\begin{matrix}\mbox{flattening} &=\frac{a-b}{a}\quad\mbox{(oblate)},\\ &=\frac{b-a}{a}\quad\mbox{(prolate)};\end{matrix}\,\!$

## First, second and third flattening

### Oblate

The first, primary flattening, f, is the versine of the spheroid's angular eccentricity ("$o\!\varepsilon\,\!$"), equaling the relative difference between its equatorial radius, $a\,\!$, and its polar radius, $b\,\!$:

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

There is also a second flattening, f' ,

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

and a third flattening[1][2], f'' (sometimes denoted as “n”, the notation of which first used in 1837 by Friedrich Bessel on calculation of meridian arc length[3]), that is the squared half-angle tangent of $o\!\varepsilon\,\!$:

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

### Prolate

With prolate valuations, the radii positions in the numerator switch, but the denominators’ remain the same, resulting in f and f' switching values of the equivalent oblate form:

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

## Numerical values for planets

For the Earth modelled by the WGS84 ellipsoid the defining values are[4]

1/f (inverse flattening): 298.257223563,

from which one derives

so that the difference of the major and minor semi-axes is about 21.385 km. (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).[5]

The amount of flattening depends on the size, density and elasticity of the celestial body (see Figure of the Earth), its rotation, and the balance of gravity and centrifugal force.