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Flattening

Measure of compression between circle to ellipse or sphere to an ellipsoid of revolution From Wikipedia, the free encyclopedia

Flattening
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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

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

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

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Definitions

There are three variants: the flattening [1] sometimes called the first flattening,[2] as well as two other "flattenings" and each sometimes called the second flattening,[3] sometimes only given a symbol,[4] or sometimes called the second flattening and third flattening, respectively.[5]

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).

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Identities

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The flattenings can be related to each-other:

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

where is the eccentricity.

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See also

References

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