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 of the resulting ellipse or ellipsoid is

A circle of radius a compressed to an ellipse.
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.

Definitions


There are three variants of flattening; when it is necessary to avoid confusion, the main flattening is called the first flattening.[1][2][3] and online web texts[4][5]

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).

(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.[6]

Identities


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

where is the eccentricity.

See also


References


  1. Maling, Derek Hylton (1992). Coordinate Systems and Map Projections (2nd ed.). Oxford; New York: Pergamon Press. ISBN 0-08-037233-3.
  2. Snyder, John P. (1987). Map Projections: A Working Manual. U.S. Geological Survey Professional Paper. 1395. Washington, D.C.: United States Government Printing Office.
  3. Torge, W. (2001). Geodesy (3rd edition). de Gruyter. ISBN 3-11-017072-8
  4. Osborne, P. (2008). The Mercator Projections Archived 2012-01-18 at the Wayback Machine Chapter 5.
  5. Rapp, Richard H. (1991). Geometric Geodesy, Part I. Dept. of Geodetic Science and Surveying, Ohio State Univ., Columbus, Ohio.
  6. F. W. Bessel, 1825, Uber die Berechnung der geographischen Langen und Breiten aus geodatischen Vermessungen, Astron.Nachr., 4(86), 241–254, doi:10.1002/asna.201011352, translated into English by C. F. F. Karney and R. E. Deakin as The calculation of longitude and latitude from geodesic measurements, Astron. Nachr. 331(8), 852–861 (2010), E-print arXiv:0908.1824, Bibcode:1825AN......4..241B