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Flattening

From Wikipedia, the free encyclopedia
Measure of compression between circle to ellipse or sphere to an ellipsoid of revolution
"Ellipticity" redirects here. For ellipticity in differential calculus, seeelliptic operator. For other uses, seeFlattening (disambiguation).
A circle of radiusa compressed to an ellipse.
A sphere of radiusa compressed to an oblate ellipsoid of revolution.

Flattening is a measure of the compression of acircle orsphere along a diameter to form anellipse or anellipsoid of revolution (spheroid) respectively. Other terms used areellipticity, oroblateness. The usual notation for flattening isf{\displaystyle f} and its definition in terms of thesemi-axesa{\displaystyle a} andb{\displaystyle b} of the resulting ellipse or ellipsoid is

f=a−ba.{\displaystyle f={\frac {a-b}{a}}.}

Thecompression factor isb/a{\displaystyle b/a} in each case; for the ellipse, this is also itsaspect ratio.

Definitions

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There are three variants: the flatteningf,{\displaystyle f,}[1] sometimes called thefirst flattening,[2] as well as two other "flattenings"f′{\displaystyle f'} andn,{\displaystyle n,} each sometimes called thesecond flattening,[3] sometimes only given a symbol,[4] or sometimes called thesecond flattening andthird flattening, respectively.[5]

In the following,a{\displaystyle a} is the larger dimension (e.g. semimajor axis), whereasb{\displaystyle b} is the smaller (semiminor axis). All flattenings are zero for a circle (a =b).

(First) flatteningf{\displaystyle f}a−ba{\displaystyle {\frac {a-b}{a}}}Fundamental. Geodeticreference ellipsoids are specified by giving1f{\displaystyle {\frac {1}{f}}\,\!}
Second flatteningf′{\displaystyle f'}a−bb{\displaystyle {\frac {a-b}{b}}}Rarely used.
Third flatteningn{\displaystyle n}a−ba+b{\displaystyle {\frac {a-b}{a+b}}}Used in geodetic calculations as a small expansion parameter.[6]

Identities

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The flattenings can be related to each-other:

f=2n1+n,n=f2−f.{\displaystyle {\begin{aligned}f={\frac {2n}{1+n}},\\[5mu]n={\frac {f}{2-f}}.\end{aligned}}}

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

ba=1−f=1−n1+n,e2=2f−f2=4n(1+n)2,f=1−1−e2,{\displaystyle {\begin{aligned}{\frac {b}{a}}&=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{aligned}}}

wheree{\displaystyle e} is theeccentricity.

See also

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References

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  1. ^Snyder, John P. (1987).Map Projections: A Working Manual. U.S. Geological Survey Professional Paper. Vol. 1395. Washington, D.C.: U.S. Government Printing Office.doi:10.3133/pp1395.
  2. ^Tenzer, Róbert (2002)."Transformation of the Geodetic Horizontal Control to Another Reference Ellipsoid".Studia Geophysica et Geodaetica.46 (1):27–32.doi:10.1023/A:1019881431482.S2CID 117114346.ProQuest 750849329.
  3. ^For example,f′{\displaystyle f'} is called thesecond flattening in:Taff, Laurence G. (1980).An Astronomical Glossary (Technical report). MIT Lincoln Lab. p. 84.
    However,n{\displaystyle n} is called thesecond flattening in:Hooijberg, Maarten (1997).Practical Geodesy: Using Computers. Springer. p. 41.doi:10.1007/978-3-642-60584-0_3.
  4. ^Maling, Derek Hylton (1992).Coordinate Systems and Map Projections (2nd ed.). Oxford; New York:Pergamon Press. p. 65.ISBN 0-08-037233-3.
    Rapp, Richard H. (1991).Geometric Geodesy, Part I (Technical report). Ohio State Univ. Dept. of Geodetic Science and Surveying.
    Osborne, P. (2008)."The Mercator Projections"(PDF). §5.2. Archived fromthe original(PDF) on 2012-01-18.
  5. ^Lapaine, Miljenko (2017). "Basics of Geodesy for Map Projections". In Lapaine, Miljenko; Usery, E. Lynn (eds.).Choosing a Map Projection. Lecture Notes in Geoinformation and Cartography. pp. 327–343.doi:10.1007/978-3-319-51835-0_13.ISBN 978-3-319-51834-3.
    Karney, Charles F.F. (2023). "On auxiliary latitudes".Survey Review:1–16.arXiv:2212.05818.doi:10.1080/00396265.2023.2217604.S2CID 254564050.
  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 asThe calculation of longitude and latitude from geodesic measurements,Astron. Nachr. 331(8), 852–861 (2010), E-printarXiv:0908.1824,Bibcode:1825AN......4..241B
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