# 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, f'' (sometimes denoted as “n”, the notation of which first used in 1837 by Friedrich Bessel on calculation of meridian arc length), 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

1/f (inverse flattening): 298.257223563,

from which one derives