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A circle of radius a compressed to an ellipse.

A sphere of radius a compressed to an oblate ellipsoid of revolution oblate.

Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening is f and its definition in terms of the semi-axes of the resulting ellipse or ellipsoid is

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

The compression factor is b/a in each case. For the ellipse, this factor is also the aspect ratio of the ellipse.

There are two other variants of flattening (see below) and when it is necessary to avoid confusion the above flattening is called the first flattening. The following definitions may be found in standard texts and online web texts

Definitions of flattening

In the following, a is the larger dimension (e.g. semimajor axis), whereas b is the smaller (semiminor axis). All flattenings are zero for a circle (a=b).

(first) flattening	$f\,\!$	${\frac {a-b}{a}}\,\!$	Fundamental. The inverse 1/f is the normal choice for geodetic reference ellipsoids.
second flattening	$f'\,\!$	${\frac {a-b}{b}}\,\!$	Rarely used.
third flattening	$n\quad (f'')\,\!$	${\frac {a-b}{a+b}}\,\!$	Used in geodetic calculations as a small expansion parameter.

Identities involving flattening

The flattenings are related to other parameters of the ellipse. For example:

${\begin{aligned}b&=a(1-f)=a\left({\frac {1-n}{1+n}}\right),\\e^{2}&=2f-f^{2}={\frac {4n}{(1+n)^{2}}}.\\\end{aligned}}$

Numerical values for planets

For the Earth modelled by the WGS84 ellipsoid the defining values are

a (equatorial radius): 6 378 137.0 m

1/f (inverse flattening): 298.257 223 563

from which one derives

b (polar radius): 6 356 752.3142 m,

so that the difference of the major and minor semi-axes is Template:Convert. (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. Because this would be virtually indistinguishable from a sphere shown as 300px by 300px, illustrations typically greatly exaggerate the flattening in cases where the image needs to represent the oblateness of the Earth.)

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). The amount of flattening depends on the density and the balance of gravitational force and centrifugal force.

See also

• Astronomy
• Earth ellipsoid
• Earth's rotation
• Eccentricity (mathematics)
• Equatorial bulge
• Gravitational field
• Gravity formula
• Ovality
• Planetology

References

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