Flattening

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

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 and its definition in terms of the semi-axes and of the resulting ellipse or ellipsoid is

The compression factor is in each case; for the ellipse, this is also its aspect ratio.

Definitions

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

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

(First) flattening  Fundamental. Geodetic reference ellipsoids are specified by giving
Second flattening Rarely used.
Third flattening  Used in geodetic calculations as a small expansion parameter.

Identities

The flattenings can be related to each-other:

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

where is the eccentricity.

See also


This page was last updated at 2023-12-06 11:06 UTC. Update now. View original page.

All our content comes from Wikipedia and under the Creative Commons Attribution-ShareAlike License.


Top

If mathematical, chemical, physical and other formulas are not displayed correctly on this page, please useFirefox or Safari