HOME

TheInfoList



Click Here for Items Related To - Flattening Factor
OR:
Flattening Factor on:   [Wikipedia]   [Google]   [Amazon]

Flattening is a measure of the compression of a
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
or
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
along a diameter to form an
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
or an
ellipsoid An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a Surface (mathemat ...
of revolution (
spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface (mathematics), surface obtained by Surface of revolution, rotating an ellipse about one of its principal axes; in other words, an ellipsoid with t ...
) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening is f and its definition in terms of the semi-axes a and b of the resulting ellipse or ellipsoid is : f =\frac . The ''compression factor'' is b/a in each case; for the ellipse, this is also its
aspect ratio The aspect ratio of a geometry, geometric shape is the ratio of its sizes in different dimensions. For example, the aspect ratio of a rectangle is the ratio of its longer side to its shorter side—the ratio of width to height, when the rectangl ...
.


Definitions

There are three variants: the flattening f, sometimes called the ''first flattening'', as well as two other "flattenings" f' and n, each sometimes called the ''second flattening'', sometimes only given a symbol, or sometimes called the ''second flattening'' and ''third flattening'', respectively. 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 (). ::


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 \frac ba &= 1-f = \frac, \\ 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 figure of the Earth, Earth's form, used as a frame of reference, reference frame for computations in geodesy, astronomy, and the geosciences. Various different ell ...
* *
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 ...
*
Ovality In telecommunications and fiber optics, ovality or noncircularity is the degree of deviation from perfect circularity of the cross section of the core or cladding of the fiber. The cross-sections of the core and cladding are assumed to a first appr ...
*
Planetary flattening A planetary coordinate system (also referred to as ''planetographic'', ''planetodetic'', or ''planetocentric'') is a generalization of the geographic, geodetic, and the geocentric coordinate systems for planets other than Earth. Similar coordi ...
*
Sphericity Sphericity is a measure of how closely the shape of a physical object resembles that of a perfect sphere. For example, the sphericity of the ball (bearing), balls inside a ball bearing determines the quality (business), quality of the bearing, ...
*
Roundness (object) Roundness is the measure of how closely the shape of an object approaches that of a mathematically perfect circle. Roundness applies in Plane (mathematics), two dimensions, such as the cross section (geometry), cross sectional circles along a cyl ...


References

{{reflist Celestial mechanics Geodesy Trigonometry Circles Ellipsoids