Flattening
f l a t t e n i n g = f = a − b a . displaystyle mathrm flattening =f= frac ab a . 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 (see below) 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[1][2][3] and online web texts[4][5] Contents 1 Definitions of flattening 2 Identities involving flattening 3 Numerical values for planets 4 Origin of flattening 5 See also 6 References Definitions of flattening[edit] 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 f displaystyle f,! a − b a displaystyle frac ab a ,! Fundamental. Geodetic reference ellipsoids are specified by giving 1/ f displaystyle f,! second flattening f ′ displaystyle f',! a − b b displaystyle frac ab b ,! Rarely used. third flattening n , ( f ″ ) displaystyle n,quad (f''),!
a − b a + b displaystyle frac ab a+b ,! Used in geodetic calculations as a small expansion parameter.[6] Identities involving flattening[edit] The flattenings are related to other parameters of the ellipse. For example: b = a ( 1 − f ) = a ( 1 − n 1 + n ) , e 2 = 2 f − f 2 = 4 n ( 1 + n ) 2 . displaystyle begin aligned b&=a(1f)=aleft( frac 1n 1+n right),\e^ 2 &=2ff^ 2 = frac 4n (1+n)^ 2 .\end aligned where e displaystyle e is the eccentricity.
Numerical values for planets[edit]
For the
WGS84
a (equatorial radius): 6 378 137.0 m 1/f (inverse flattening): 298.257 223 563 from which one derives b (polar radius): 6 356 752.3142 m, so that the difference of the major and minor semiaxes is
21.385 km (13 mi). (This is only 0.335% of the major
axis, so a representation of
Earth
Astronomy
Earth
References[edit] ^ Maling, Derek Hylton (1992). Coordinate Systems and Map Projections
(2nd ed.). Oxford; New York: Pergamon Press.
ISBN 0080372333.
^ Snyder, John P. (1987). Map Projections: A Working Manual. U.S.
Geological Survey Professional Paper. 1395. Washington, D.C.: United
States Government Printing Office.
^ Torge, W. (2001). Geodesy (3rd edition). de Gruyter.
ISBN 3110170728
^ Osborne, P. (2008). The Mercator Projections Archived 20120118 at
the Wayback Machine. Chapter 5.
^ Rapp, Richard H. (1991). Geometric Geodesy, Part I. Dept. of
Geodetic Science and Surveying, Ohio State Univ., Columbus, Ohio. [1]
^ F. W. Bessel, 1825, Uber die Berechnung der geographischen Langen
und Breiten aus geodatischen Vermessungen, Astron.Nachr., 4(86),
241254, 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), 852861
(2010), Eprint arXiv:0908.1824, Bibcode: 1825AN......4..241B
^ The
WGS84
