Flattening

Measure of compression between circle to ellipse or sphere to an ellipsoid of revolution
A circle of radius a compressed to an ellipse.
A sphere of radius a compressed to an oblate ellipsoid of revolution.

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 {\displaystyle f} and its definition in terms of the semi-axes a {\displaystyle a} and b {\displaystyle b} of the resulting ellipse or ellipsoid is

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

The compression factor is b / a {\displaystyle b/a} in each case; for the ellipse, this is also its aspect ratio.

Definitions

There are three variants: the flattening f , {\displaystyle f,} [1] sometimes called the first flattening,[2] as well as two other "flattenings" f ′ {\displaystyle f'} and n , {\displaystyle n,} 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, a {\displaystyle a} is the larger dimension (e.g. semimajor axis), whereas b {\displaystyle 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 {a-b}{a}}} Fundamental. Geodetic reference ellipsoids are specified by giving 1 f {\displaystyle {\frac {1}{f}}\,\!}
Second flattening f ′ {\displaystyle f'} a − b b {\displaystyle {\frac {a-b}{b}}} Rarely used.
Third flattening  n {\displaystyle n} a − b a + b {\displaystyle {\frac {a-b}{a+b}}} Used in geodetic calculations as a small expansion parameter.[6]

Identities

The flattenings can be related to each-other:

f = 2 n 1 + n , n = f 2 − 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,

b a = 1 − f = 1 − n 1 + n , e 2 = 2 f − f 2 = 4 n ( 1 + n ) 2 , f = 1 − 1 − e 2 , {\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}}}

where e {\displaystyle e} is the eccentricity.

See also

  • Earth flattening
  • Eccentricity (mathematics) § Ellipses
  • Equatorial bulge
  • Ovality
  • Planetary flattening
  • Sphericity
  • Roundness (object)

References

  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 the second flattening in: Taff, Laurence G. (1980). An Astronomical Glossary (Technical report). MIT Lincoln Lab. p. 84.
    However, n {\displaystyle n} is called the second 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 from the 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 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