Flattening

There are three variants: the flattening $f,${{cite book |author=Snyder, John P. |title=Map Projections: A Working Manual |series=U.S. Geological Survey Professional Paper |volume=1395 |year=1987 |publisher=U.S. Government Printing Office |location=Washington, D.C. |doi=10.3133/pp1395 |doi-access=free |url=https://pubs.er.usgs.gov/publication/pp1395 }} sometimes called the first flattening,

{{cite journal |last=Tenzer |first=Róbert |title=Transformation of the Geodetic Horizontal Control to Another Reference Ellipsoid |journal=Studia Geophysica et Geodaetica |volume=46 |issue=1 |year=2002 |pages=27–32 |doi=10.1023/A:1019881431482 |s2cid=117114346 |url=https://www.proquest.com/docview/750849329|id={{ProQuest|750849329}} }} as well as two other "flattenings" $f\text{'}$ and $n,$ each sometimes called the second flattening,For example, $f\text{'}$ is called the second flattening in: {{cite tech report |last=Taff |first=Laurence G. |title=An Astronomical Glossary |publisher=MIT Lincoln Lab |year=1980 |url=https://apps.dtic.mil/sti/citations/ADA084445 |page=84}} {{pb}} However, $n$ is called the second flattening in: {{cite book |last=Hooijberg |first=Maarten |title=Practical Geodesy: Using Computers |page=41 |publisher=Springer |year=1997 |doi=10.1007/978-3-642-60584-0_3}} sometimes only given a symbol,{{cite book | last=Maling |first=Derek Hylton | title=Coordinate Systems and Map Projections |edition=2nd |year=1992 | publisher =Pergamon Press|location=Oxford; New York |isbn=0-08-037233-3 |page=65}} {{pb}} {{cite tech report |last=Rapp |first=Richard H. |year=1991 |title=Geometric Geodesy, Part I |publisher=Ohio State Univ. Dept. of Geodetic Science and Surveying |url=http://hdl.handle.net/1811/24333}} {{pb}} {{cite web |last=Osborne |first=P. |year=2008 |title=The Mercator Projections |url=http://mercator.myzen.co.uk/mercator.pdf |url-status=dead |archive-url=https://web.archive.org/web/20120118224152/http://mercator.myzen.co.uk/mercator.pdf |archive-date=2012-01-18 |at=§5.2 }} or sometimes called the second flattening and third flattening, respectively.{{cite book |last=Lapaine |first=Miljenko |chapter=Basics of Geodesy for Map Projections |editor1-last=Lapaine |editor1-first=Miljenko |editor2-last=Usery |editor2-first=E. Lynn |title=Choosing a Map Projection |series=Lecture Notes in Geoinformation and Cartography |year=2017 |pages=327–343 |doi=10.1007/978-3-319-51835-0_13|isbn=978-3-319-51834-3 }}{{pb}}{{cite journal |last=Karney |first=Charles F.F. |year=2023 |title=On auxiliary latitudes |journal=Survey Review |pages=1–16 |doi=10.1080/00396265.2023.2217604 |arxiv=2212.05818|s2cid=254564050 }}

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 ({{math|a {{=}} b}}).

::

The flattenings can be related to each-other:

:$\backslash begin\{align\}$

f = \frac{2n}{1 + n}, \\[5mu]

n = \frac{f}{2 - f}.

\end{align}

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

:$\backslash begin\{align\}$

\frac ba &= 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{align}

where $e$ is the eccentricity.

• Earth flattening
• {{section link|Eccentricity (mathematics)|Ellipses}}
• Equatorial bulge
• Ovality
• Planetary flattening
• Sphericity
• Roundness (object)

{{reflist}}

Category:Celestial mechanics

Category:Geodesy

Category:Trigonometry

Category:Circles

Category:Ellipsoids