Flattening


Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution 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
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 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, whereas b is the smaller. All flattenings are zero for a circle.

Identities involving flattening

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

Numerical values for planets

For the WGS84 ellipsoid to model Earth, the defining values are
from which one derives
so that the difference of the major and minor semi-axes is.
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 about.

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