Flattening

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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 — 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

• Flattening — Рихтовка (стереотипа); Потеря контраста; Смягчение (изображения) …   Краткий толковый словарь по полиграфии

• flattening — flat·ten·ing (flatґə ning) making flat; diminishing …   Medical dictionary

• flattening — Смотри Плющение …   Энциклопедический словарь по металлургии

• flattening — n. act of leveling, straightening, making flat flatÂ·ten || flÃ¦tn v. make level or smooth; become level; make flat; beat flat; knock down; make insipid or dull …   English contemporary dictionary

• flattening — …   Useful english dictionary

• 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