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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 ^{[1] [2][3]} and online web texts^[4][5]

Definitions of flattening[link]

(first) flattening	Failed to parse (Missing texvc executable; please see math/README to configure.): f\,\!	Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{a-b}{a}\,\!	Fundamental. The inverse 1/f is the normal choice for geodetic reference ellipsoids.
second flattening	Failed to parse (Missing texvc executable; please see math/README to configure.): f'\,\!	Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{a-b}{b}\,\!	Rarely used.
third flattening	Failed to parse (Missing texvc executable; please see math/README to configure.): n\quad(f'')\,\!	Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{a-b}{a+b}\,\!	Used in geodetic calculations as a small expansion parameter.[6]

Identities involving flattening[link]

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} b&=a(1-f)=a\left(\frac{1-n}{1+n}\right),\\ e^2&=2f-f^2 = \frac{4n}{(1+n)^2}.\\ \end{align}

Numerical values for planets[link]

For the Earth modelled by the WGS84 ellipsoid the defining values are^[7]

a (equatorial radius): 6,378,137.0 m

1/f (inverse flattening): 298.257223563

from which one derives

b (polar radius): 6,356,752.3142 m,

so that the difference of the major and minor semi-axes is 21.385 km (13 mi). (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.^{[citation needed]})

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[link]

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).^[8] The amount of flattening depends on the density and the balance of gravitational force and centrifugal force.

See also[link]

• Earth ellipsoid,
• Equatorial bulge, Earth's rotation
• Astronomy
• Planetology
• Gravitational field
• Gravity formula

References[link]

Sean Reinert
Sean Reinert
Background information
Born	(1971-05-27) May 27, 1971 (age 41)
Genres	Progressive metal, technical metal
Occupations	Musician, songwriter
Instruments	Drums, keyboards, percussion, tabla, vocals
Years active	1987−present
Associated acts	Cynic, Æon Spoke, Death, Gordian Knot, Aghora

Sean Reinert (born May 27, 1971) is the drummer of Cynic and Æon Spoke. He is known for his technical, original and creative drumming technique.

He credits such drummers as Dave Miranda, John Bonham, Dave Weckl, Gary Husband and Vinnie Colaiuta as being huge influences in his formative years.

In 1991, Reinert and Paul Masvidal (also of Cynic) joined the band Death to record the Human album. This album is seen by many as the start of the technical death metal genre. After touring with Death, he returned to Cynic.

1993 saw the release of Focus (regarded as a landmark release in the field of technical/progressive metal) on the Roadrunner Records label. Roadrunner re-released it in 2005 as a special collector's edition due to high demand. After Cynic's disbandment Reinert had been a member of the Cynic spin-off band Portal which released a demo but eventually disbanded. Having relocated to Los Angelos, Reinert continued working with music by writing and performing for television shows and motion pictures.

In 1999, Reinert helped form Aeon Spoke alongside Masvidal, which is described as an "ethereal rock" band.^[1] The band saw releases in 2004 and 2007, receiving national and satellite airplay, performing, hosting radio appearances, and even appearing on television show and motion picture soundtracks.

During this time in 2006, it was announced that Cynic would reform for a tour which took place in the summer of 2007. The reformed line-up then recorded a follow-up to Focus entitled Traced in Air, which was released in 2008 and followed by more touring. In 2009, a new EP with Cynic was recorded scheduled for release in May, 2010.

Discography[link]

• Death − Human (1991)
• Cynic − Focus (1993)
• Sean Malone − Cortlandt (1996)
• Gordian Knot − Gordian Knot (1999), Emergent (2003)
• Æon Spoke − Above the Buried Cry (2004), Æon Spoke (2007)
• Aghora − Aghora (2000), Formless (2006)
• C-187 − Collision (2007)
• Cynic - Traced in Air (2008)
• Levi/Werstler - Avalanche of Worms (2010)
• Cynic - Re-Traced (2010)
• Cynic - Carbon-Based Anatomy (2011)

References[link]

External links[link]

• Sean Reinert on Myspace
• Official Æon Spoke website
• Official Cynic website
• Death + Sean Reinert Drum Cam - Flattening of Emotions 10.26.91

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