Editing Transverse Mercator projection (section)

===Implementations of the Gauss–Krüger projection===
In his 1912<ref name="kruger"/> paper, Krüger presented two distinct solutions, distinguished here by the expansion parameter:
* '''Krüger&ndash;''n''''' (paragraphs 5 to 8): Formulae for the direct projection, giving the coordinates ''x'' and ''y'', are fourth order expansions in terms of the third flattening, ''n'' (the ratio of the difference and sum of the major and minor axes of the ellipsoid). The coefficients are expressed in terms of latitude (''φ''), longitude (''λ''), major axis (''a'') and eccentricity (''e''). The inverse formulae for ''φ'' and ''λ'' are also fourth order expansions in ''n'' but with coefficients expressed in terms of ''x'', ''y'', ''a'' and ''e''.
* '''Krüger–''λ''''' (paragraphs 13 and 14): Formulae giving the projection coordinates ''x'' and ''y'' are expansions (of orders 5 and 4 respectively) in terms of the longitude ''λ'', expressed in radians: the coefficients are expressed in terms of ''φ'', ''a'' and ''e''. The inverse projection for ''φ'' and ''λ'' are sixth order expansions in terms of the ratio {{sfrac|''x''|''a''}}, with coefficients expressed in terms of ''y'', ''a'' and ''e''. (See [[Transverse Mercator: Redfearn series]].)

The Krüger–''λ'' series were the first to be implemented, possibly because they were much easier to evaluate on the hand calculators of the mid twentieth century.
*'''Lee–Redfearn–OSGB''': In 1945, L. P. Lee<ref name=lee_series>[[Laurence Patrick Lee|Lee, L. P.]] (1945). Survey Review, Volume&nbsp;'''8''' (Part&nbsp;58), pp&nbsp;142–152. [http://www.ingentaconnect.com/content/maney/sre/1945/00000008/00000058/art00004 ''The transverse Mercator projection of the spheroid'']. (Errata and comments in Volume&nbsp;'''8''' (Part&nbsp;61), pp.&nbsp;277–278.</ref> confirmed the ''λ'' expansions of Krüger and proposed their adoption by the OSGB<ref name=osgb>A guide to coordinate systems in Great Britain. This is available as a pdf document at
{{cite web |url=http://www.ordnancesurvey.co.uk/oswebsite/gps/information/coordinatesystemsinfo/guidecontents |title=Welcome to GPS Network |access-date=2012-01-11 |url-status=dead |archive-url=https://web.archive.org/web/20120211075826/http://www.ordnancesurvey.co.uk/oswebsite/gps/information/coordinatesystemsinfo/guidecontents/ |archive-date=2012-02-11 }}</ref> but Redfearn (1948)<ref name=Redfearn>Redfearn, J C B (1948). Survey Review, Volume&nbsp;'''9''' (Part&nbsp;69), pp&nbsp;318–322, [http://www.ingentaconnect.com/content/maney/sre/1948/00000009/00000069/art00005 ''Transverse Mercator formulae''].</ref> pointed out that they were not accurate because of (a) the relatively high latitudes of Great Britain and (b) the great width of the area mapped, over 10 degrees of longitude. Redfearn extended the series to eighth order and examined which terms were necessary to attain an accuracy of 1&nbsp;mm (ground measurement). The [[Transverse Mercator: Redfearn series|Redfearn series]] are still the basis of the OSGB map projections.<ref name=osgb />
*'''Thomas–UTM''': The ''λ'' expansions of Krüger were also confirmed by Paul Thomas in 1952:<ref>Thomas, Paul D (1952). ''Conformal Projections in Geodesy and Cartography''. Washington: U.S. Coast and Geodetic Survey Special Publication 251.</ref> they are readily available in Snyder.<ref name=snyder /> His projection formulae, completely equivalent to those presented by Redfearn, were adopted by the United States Defence Mapping Agency as the basis for the [[Universal Transverse Mercator coordinate system|UTM]].<ref name=utm>{{cite journal |first1=J. W. |last1=Hager |first2=J. F. |last2=Behensky |first3=B. W. |last3=Drew |year=1989 |title=The universal grids: Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS) |journal=Technical Report TM 8358.2, Defense Mapping Agency |url=http://earth-info.nga.mil/GandG/publications/tm8358.2/TM8358_2.pdf }}</ref> They are also incorporated into the GEOTRANS<ref>{{Cite web | year=2024 | title= Office of Geomatics | url=https://earth-info.nga.mil/index.php}}</ref> coordinate converter made available by the United States National Geospatial-Intelligence Agency’s Office of Geomatics.
*'''Other countries''': The Redfearn series are the basis for geodetic mapping in many countries: Australia, Germany, Canada, South Africa to name but a few. (A list is given in Appendix A.1 of Stuifbergen 2009.)<ref>{{Cite journal | author=N. Stuifbergen | year=2009 | title=Wide zone transverse Mercator projection | journal=Canadian Technical Report of Hydrography and Ocean Sciences| issue=262 | publisher=Canadian Hydrographic Service | url=http://www.dfo-mpo.gc.ca/Library/337182.pdf | archive-url=https://web.archive.org/web/20160809191021/http://www.dfo-mpo.gc.ca/Library/337182.pdf | archive-date=2016-08-09}}</ref>
*Many variants of the Redfearn series have been proposed but only those adopted by national cartographic agencies are of importance. For an example of modifications which do not have this status see [[Transverse Mercator: Bowring series]]). All such modifications have been eclipsed by the power of modern computers and the development of high order ''n''-series outlined below.  The precise Redfearn series, although of low order, cannot be disregarded as they are still enshrined in the quasi-legal definitions of OSGB and UTM etc.

The Krüger–''n'' series have been implemented (to fourth order in ''n'') by the following nations.
*France<ref>{{Cite web | url=http://geodesie.ign.fr/contenu/fichiers/documentation/algorithmes/notice/NTG_76.pdf | access-date=2024-07-27 | title=Projection Cartographique Mercator Traverse | language=fr | website=geodesie.ign.fr | date=January 1995 | publisher=[[Institut Geographique National]]}}</ref>
*Finland<ref>{{Cite web | author1=R. Kuittinen | author2=T. Sarjakoski | author3=M. Ollikainen| author4= M. Poutanen| author5= R. Nuuros| author6= P. Tätilä| author7= J. Peltola| author8=R. Ruotsalainen| author9= M. Ollikainen | year=2006 | title=ETRS89—järjestelmään liittyvät karttaprojektiot, tasokoordinaatistot ja karttalehtijako, Liite 1: Projektiokaavat | language=fi|  journal=Technical Report JHS | issue=154 | publisher=Finnish Geodetic Institute | url=http://docs.jhs-suositukset.fi/jhs-suositukset/JHS154/JHS154_liite1.pdf | trans-title=map projections related to the ETRS89 system, level coordinates and map sheet division, Appendix 1: Project formulas}}</ref>
*Sweden<ref>{{Cite web |url=https://www.lantmateriet.se/globalassets/geodata/gps-och-geodetisk-matning/gauss_conformal_projection.pdf |title=Gauss Conformal Projection  (Transverse Mercator): Krüger’s Formulas |access-date=2024-07-27}}</ref>
*Japan<ref>{{Cite web | url=http://psgsv2.gsi.go.jp/koukyou/jyunsoku/pdf/H28/H28_junsoku_furoku6.pdf#page=22 | page=22 | title=座標を変換して経緯度、子午線収差角及び縮尺係数を求める計算 | language=zh | trans-title=Calculation to convert coordinates to obtain longitude and latitude, meridian aberration angle and scale factor | archive-url=https://web.archive.org/web/20180508054725/http://psgsv2.gsi.go.jp/koukyou/jyunsoku/pdf/H28/H28_junsoku_furoku6.pdf#page=22 | archive-date=2018-05-08}}</ref>

Higher order versions of the Krüger–''n'' series have been implemented to seventh order by Engsager and Poder<ref name=poder>K. E. Engsager and K. Poder, 2007, [http://icaci.org/documents/ICC_proceedings/ICC2007/documents/doc/THEME%202/oral%201/2.1.2%20A%20HIGHLY%20ACCURATE%20WORLD%20WIDE%20ALGORITHM%20FOR%20THE%20TRANSVE.doc A highly accurate world wide algorithm for the transverse Mercator mapping (almost)], in Proc. XXIII Intl. Cartographic Conf. (ICC2007), Moscow, p. 2.1.2.</ref> and to tenth order by Kawase.<ref name=kawase>Kawase, K. (2011): [http://www.gsi.go.jp/common/000062452.pdf A General Formula for Calculating Meridian Arc Length and its Application to Coordinate Conversion in the Gauss–Krüger Projection], Bulletin of the [[Geospatial Information Authority of Japan]], '''59''', pp&nbsp;1–13</ref> Apart from a series expansion for the transformation between latitude and conformal latitude, Karney has implemented the series to thirtieth order.<ref name=karney />