# Flattening

Flattening

The flattening, ellipticity, or oblateness of an oblate spheroid 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

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

## Definition of flattening

### First order

The first, primary flattening, ƒ, is the versine of the spheroid's angular eccentricity α, equalling the relative difference between its equatorial radius, a, and its polar radius, b:

$f=\mathrm{ver}(\alpha)=2\sin^2\left(\frac{\alpha}{2}\right)=1-\cos(\alpha)=\frac{a-b}{a};\,\!$

### Second and third orders

There is also a second flattening, f' ,

$f'=\frac{2\sin^2(\alpha/2)}{1-2\sin^2(\alpha/2)}=\frac{a-b}{b}$

and a third flattening,[1][2] f' ' (sometimes denoted as "n" – a notation first used in 1837 by Friedrich Bessel on calculation of meridian arc length[3] – that is the squared half-angle tangent of α:

$f''=n=\tan^2\left(\frac{\alpha}{2}\right)=\frac{1-\cos(\alpha)}{1+\cos(\alpha)}=\frac{a-b}{a+b};\,\!$

## First order flattening of planets

• The flattening of the smoothed Earth's surface in the World Geodetic System (WGS-84) is 1:298.257223563 (which corresponds to a radius difference of 21.385 km (13 mi) 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

and in detail on

## References

1. ^ 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
2. ^ Ганьшин, В. Н. (1967): Геометрия земного эллипсоида, Издательство «Недра», Москва
3. ^ 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

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### Look at other dictionaries:

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• Flattening — Flatten Flat ten (fl[a^]t t n), v. t. [imp. & p. p. {Flattened}; p. pr. & vb. n. {Flattening}.] [From {Flat}, a.] 1. To reduce to an even surface or one approaching evenness; to make flat; to level; to make plane. [1913 Webster] 2. To throw down; …   The Collaborative International Dictionary of English

• Flattening dies — Flattening dies. См. Правильные штампы. (Источник: «Металлы и сплавы. Справочник.» Под редакцией Ю.П. Солнцева; НПО Профессионал , НПО Мир и семья ; Санкт Петербург, 2003 г.) …   Словарь металлургических терминов

• Flattening test — Flattening test. См. Правильное испытание. (Источник: «Металлы и сплавы. Справочник.» Под редакцией Ю.П. Солнцева; НПО Профессионал , НПО Мир и семья ; Санкт Петербург, 2003 г.) …   Словарь металлургических терминов

• Flattening oven — Flatten Flat ten (fl[a^]t t n), v. t. [imp. & p. p. {Flattened}; p. pr. & vb. n. {Flattening}.] [From {Flat}, a.] 1. To reduce to an even surface or one approaching evenness; to make flat; to level; to make plane. [1913 Webster] 2. To throw down; …   The Collaborative International Dictionary of English

• flattening out — UK US flatten off/out Phrasal Verb with flatten({{}}/ˈflætən/ verb [I or T] ► to stop rising or falling, and stay at the same level: »If the housing market continues to flatten out or even decline, we re in for some tough times.… …   Financial and business terms

• flattening oven — noun Etymology: flattening from gerund of flatten : a heating chamber in which split glass cylinders are flattened and annealed …   Useful english dictionary

• flattening stone — noun : a stone used in a flattening oven …   Useful english dictionary

• flattening — noun a) The act, or the result of making something flat of flatter The Earth is a sphere that has flattenings at the poles. b) A flattened part of something …   Wiktionary