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Concept
Flattening
Summary
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.
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).
{| class="wikitable" style="border:1px solid darkgray;" cellpadding="5"
! style="padding-left: 0.5em" scope="row" | (First) flattening
| style="padding-left: 0.5em" |
| style="padding-left: 0.5em" |
| style="padding-left: 0.5em " | Fundamental. Geodetic reference ellipsoids are specified by giving
|-
! style="padding-left: 0.5em" scope="row" | Second flattening
| style="padding-left: 0.5em" |
| style="padding-left: 0.5em" |
| style="padding-left: 0.5em" | Rarely used.
|-
! style="padding-left: 0.5em" scope="row" | Third flattening
| style="padding-left: 0.5em" |
| style="padding-left: 0.5em" |
| style="padding-left: 0.5em" | Used in geodetic calculations as a small expansion parameter.
|}
The flattenings can be related to each-other:
The flattenings are related to other parameters of the ellipse. For example,
where is the eccentricity.
Official source
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