Flattening
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Flattening is a measure of the compression of a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
or
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
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 and its definition in terms of the semi-axes of the resulting ellipse or ellipsoid is :: \mathrm = f =\frac . The ''compression factor'' is \frac\,\! in each case; for the ellipse, this is also its aspect ratio.


Definitions

There are three variants of flattening; when it is necessary to avoid confusion, the main flattening is called the first flattening.Torge, W. (2001). ''Geodesy'' (3rd edition). de Gruyter. and online web textsOsborne, P. (2008).
The Mercator Projections
'' Chapter 5.Rapp, Richard H. (1991). ''Geometric Geodesy, Part I''. Dept. of Geodetic Science and Surveying, Ohio State Univ., Columbus, Ohio

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


Identities

The flattenings can be related to each-other: :\begin f = \frac, \\ mun = \frac. \end The flattenings are related to other parameters of the ellipse. For example, :\begin b &= a(1-f) = a\left(\frac\right), \\ mue^2 &= 2f-f^2 = \frac, \\ muf &= 1-\sqrt, \end where e is the
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
.


See also

*
Earth flattening An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximations ...
* *
Equatorial bulge An equatorial bulge is a difference between the equatorial and polar diameters of a planet, due to the centrifugal force exerted by the rotation about the body's axis. A rotating body tends to form an oblate spheroid rather than a sphere. On E ...
* Ovality *
Planetary flattening A planetary coordinate system is a generalization of the geographic coordinate system and the geocentric coordinate system for planets other than Earth. Similar coordinate systems are defined for other solid celestial bodies, such as in the '' sel ...
*
Sphericity Sphericity is a measure of how closely the shape of an object resembles that of a perfect sphere. For example, the sphericity of the balls inside a ball bearing determines the quality of the bearing, such as the load it can bear or the speed a ...
*
Roundness (object) Roundness is the measure of how closely the shape of an object approaches that of a mathematically perfect circle. Roundness applies in two dimensions, such as the cross sectional circles along a cylindrical object such as a shaft or a cylindri ...


References

{{reflist Celestial mechanics Geodesy Trigonometry Circles