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 (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 $$a$$ and $$b$$ of the resulting ellipse or ellipsoid is
 * $$ f =\frac {a - b}{a}.$$

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

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 ($a = b$).
 * {| class="wikitable" style="border:1px solid darkgray;" cellpadding="5"

! style="padding-left: 0.5em" scope="row" | (First) flattening ! style="padding-left: 0.5em" scope="row" | Second flattening ! style="padding-left: 0.5em" scope="row" | Third flattening&emsp;
 * style="padding-left: 0.5em" | $$f$$
 * style="padding-left: 0.5em" | $$\frac{a-b}{a}$$
 * style="padding-left: 0.5em " | Fundamental. Geodetic reference ellipsoids are specified by giving $$\frac{1}{f}\,\!$$
 * style="padding-left: 0.5em" | $$f'$$
 * style="padding-left: 0.5em" | $$\frac{a-b}{b}$$
 * style="padding-left: 0.5em" | Rarely used.
 * style="padding-left: 0.5em" | $$n$$
 * style="padding-left: 0.5em" | $$\frac{a-b}{a+b}$$
 * style="padding-left: 0.5em" | Used in geodetic calculations as a small expansion parameter.
 * }

Identities
The flattenings can be related to each-other:


 * $$\begin{align}

f = \frac{2n}{1 + n}, \\[5mu] n = \frac{f}{2 - f}. \end{align}$$

The flattenings are related to other parameters of the ellipse. For example,
 * $$\begin{align}

\frac ba &= 1-f = \frac{1-n}{1+n}, \\[5mu] e^2 &= 2f-f^2 = \frac{4n}{(1+n)^2}, \\[5mu] f &= 1-\sqrt{1-e^2}, \end{align}$$

where $$e$$ is the eccentricity.