Editing Transverse Mercator projection (section)

==Ellipsoidal transverse Mercator==
The ellipsoidal form of the transverse Mercator projection was developed by [[Carl Friedrich Gauss]] in 1822<ref name=gauss1825>Gauss, Karl Friedrich, 1825. "Allgemeine Auflösung der Aufgabe: die Theile einer gegebnen Fläche auf einer andern gegebnen Fläche so abzubilden, daß die Abbildung dem Abgebildeten in den kleinsten Theilen ähnlich wird" Preisarbeit der Kopenhagener Akademie 1822. [https://books.google.com/books?id=q2I_AAAAcAAJ Schumacher Astronomische Abhandlungen, Altona, no. 3], p. 5&ndash;30. [Reprinted, 1894, Ostwald's Klassiker der Exakten Wissenschaften, no. 55: Leipzig, Wilhelm Engelmann, p. 57&ndash;81, with editing by Albert Wangerin, pp. 97&ndash;101. Also in Herausgegeben von der Gesellschaft der Wissenschaften zu Göttingen in Kommission bei Julius Springer in Berlin, 1929, v. 12, pp. 1&ndash;9.]</ref> and further analysed by [[Johann Heinrich Louis Krüger]] in 1912.<ref name=kruger>Krüger, L. (1912). ''[https://dx.doi.org/10.2312/GFZ.b103-krueger28 Konforme Abbildung des Erdellipsoids in der Ebene]''. Royal Prussian Geodetic Institute, New Series 52.</ref>

{{anchor|Terminology}}The projection is known by several names: the ''(ellipsoidal) transverse Mercator'' in the US; '''Gauss conformal''' or '''Gauss–Krüger''' in Europe; or '''Gauss–Krüger transverse Mercator''' more generally. 
Other than just a synonym for the ellipsoidal transverse Mercator map projection, the term '''Gauss–Krüger''' may be used in other slightly different ways:
* Sometimes, the term is used for a particular computational method for transverse Mercator: that is, how to convert between latitude/longitude and projected coordinates.  There is no simple closed formula to do so when the earth is modelled as an ellipsoid.  But the ''Gauss–Krüger'' method gives the same results as other methods, at least if you are sufficiently near the central meridian: less than 100 degrees of longitude, say.  Further away, some methods become inaccurate.
* The term is also used for a particular set of transverse Mercator projections used in narrow zones in Europe and South America, at least in Germany, Turkey, Austria, Slovenia, Croatia, Bosnia-Herzegovina, Serbia, Montenegro, North Macedonia, Finland and Argentina.  This ''Gauss–Krüger'' system is similar to the [[universal transverse Mercator]] system, but the central meridians of the Gauss–Krüger zones are only 3° apart, as opposed to 6° in UTM.

The projection is conformal with a constant scale on the central meridian. (There are other conformal generalisations of the transverse Mercator from the sphere to the ellipsoid but only Gauss-Krüger has a constant scale on the central meridian.) Throughout the twentieth century the Gauss–Krüger transverse Mercator was adopted, in one form or another, by many nations (and international bodies);<ref>{{cite web | url = http://eusoils.jrc.ec.europa.eu/Projects/Alpsis/Docs/ref_grid_sh_proc_draft.pdf | title = Short Proceedings of the 1st European Workshop on Reference Grids, Ispra, 27–29 October 2003 | publisher = [[European Environment Agency]] | date = 2004-06-14 | access-date = 2009-08-27 | page = 6}} The EEA recommends the transverse Mercator for conformal pan-European mapping at scales larger than 1:500,000.</ref> in addition it provides the basis for the [[Universal Transverse Mercator]] series of projections. The Gauss–Krüger projection is now the most widely used projection in accurate large-scale mapping.{{citation needed|date=February 2017}}

The projection, as developed by Gauss and Krüger, was expressed in terms of low order [[power series]] which were assumed to diverge in the east-west direction, exactly as in the spherical version. This was proved to be untrue by British cartographer E. H. Thompson, whose unpublished exact (closed form) version of the projection, reported by [[Laurence Patrick Lee]] in 1976,<ref name="lee_exact">{{cite book | last = Lee | first = L. P. | author-link = Laurence Patrick Lee | year = 1976 | title = Conformal Projections Based on Elliptic Functions | location = Toronto | publisher = B. V. Gutsell, York University | series = ''Cartographica Monographs'' | volume = 16 | url = https://archive.org/details/conformalproject0000leel | url-access = limited | isbn = 0-919870-16-3 }} Supplement No. 1 to [https://www.utpjournals.press/toc/cart/13/1 ''The Canadian Cartographer'' '''13''']. pp. [https://archive.org/details/conformalproject0000leel/page/n8/ 1–14], [https://archive.org/details/conformalproject0000leel/page/92/ 92–101], and [https://archive.org/details/conformalproject0000leel/page/107/ 107–114].</ref> showed that the ellipsoidal projection is finite (below). This is the most striking difference between the spherical and ellipsoidal versions of the transverse Mercator projection: Gauss–Krüger gives a reasonable projection of the ''whole'' ellipsoid to the plane, although its principal application is to accurate large-scale mapping "close" to the central meridian.{{citation needed|date=February 2017}}

[[File:MercTranEll.png|center|thumb|350px|Ellipsoidal transverse Mercator: a finite projection.]]

===Features===
*Near the central meridian (Greenwich in the above example) the projection has low distortion and the shapes of Africa, western Europe, the British Isles, Greenland, and Antarctica compare favourably with a globe.
* The central regions of the transverse projections on sphere and ellipsoid are indistinguishable on the small-scale projections shown here.
*The meridians at 90° east and west of the chosen central meridian project to horizontal lines through the poles. The more distant hemisphere is projected above the north pole and below the south pole.
*The equator bisects Africa, crosses South America and then continues onto the complete outer boundary of the projection; the top and bottom edges and the right and left edges must be identified (i.e. they represent the same lines on the globe). (Indonesia is bisected.)
*Distortion increases towards the right and left boundaries of the projection but it does not increase to infinity. Note the Galapagos Islands where the 90° west meridian meets the equator at bottom left.
*The map is conformal. Lines intersecting at any specified angle on the ellipsoid project into lines intersecting at the same angle on the projection. In particular parallels and meridians intersect at 90°.
*The point scale factor is independent of direction at any point so that the shape of a ''small'' region is reasonably well preserved. The necessary condition is that the magnitude of scale factor must not vary too much over the region concerned. Note that while South America is distorted greatly the island of Ceylon is small enough to be reasonably shaped although it is far from the central meridian.
*The choice of central meridian greatly affects the appearance of the projection. If 90°W is chosen then the whole of the Americas is reasonable. If 145°E is chosen the Far East is good and Australia is oriented with north up.

In most applications the [[Gauss–Krüger coordinate system]] is applied to a narrow strip near the central meridians where the differences between the spherical and ellipsoidal versions are small, but nevertheless important in accurate mapping. Direct series for scale, convergence and distortion are functions of eccentricity and both latitude and longitude on the ellipsoid: inverse series are functions of [[Eccentricity (mathematics)#Ellipses|eccentricity]] and both ''x'' and ''y'' on the projection. In the secant version the lines of true scale on the projection are no longer parallel to central meridian; they curve slightly. The convergence angle between projected meridians and the ''x'' constant grid lines is no longer zero (except on the equator) so that a grid bearing must be corrected to obtain an azimuth from true north. The difference is small, but not negligible, particularly at high latitudes.

===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 />

===Exact Gauss–Krüger and accuracy of the truncated series===
An exact solution by E. H. Thompson is described by L. P. Lee.<ref name=lee_exact /> It is constructed in terms of elliptic functions (defined in chapters 19 and 22 of the NIST<ref>{{Cite web | author1=F. W.J. Olver| author2=D.W. Lozier| author3=R.F. Boisvert| author4=C.W. Clark | year=2010 | title=NIST Handbook of Mathematical Functions | publisher=Cambridge University Press | url=http://dlmf.nist.gov}}</ref> handbook) which can be calculated to arbitrary accuracy using algebraic computing systems such as Maxima.<ref>{{Cite web | year=2009 | title=Maxima - A computer algebra system | url=http://maxima.sourceforge.io | website=maxima.sourceforge.io | access-date=2024-07-27}}</ref> Such an implementation of the exact solution is described by Karney (2011).<ref name=karney>{{Cite journal | author=C. F. F. Karney | year=2011 | doi=10.1007/s00190-011-0445-3 | title=Transverse Mercator with an accuracy of a few nanometers | url=https://link.springer.com/article/10.1007/s00190-011-0445-3 | journal=Journal of Geodesy | volume=85 | pages=475-485| arxiv=1002.1417 }}</ref><ref>{{Cite web | title=Transverse Mercator Projection - preprint of paper and C++ implementation of algorithms | url= https://geographiclib.sourceforge.io/tm.html | website=geographiclib.sourceforge.io}}</ref>

The exact solution is a valuable tool in assessing the accuracy of the truncated ''n'' and λ series. For example, the original 1912 Krüger–''n'' series compares very favourably with the exact values: they differ by less than 0.31&nbsp;μm within 1000&nbsp;km of the central meridian and by less than 1&nbsp;mm out to 6000&nbsp;km. On the other hand, the difference of the Redfearn series used by GEOTRANS and the exact solution is less than 1&nbsp;mm out to a longitude difference of 3 degrees, corresponding to a distance of 334&nbsp;km from the central meridian at the equator but a mere 35&nbsp;km at the northern limit of an UTM zone. Thus the Krüger–''n'' series are very much better than the Redfearn λ series.

The Redfearn series becomes much worse as the zone widens. Karney discusses Greenland as an instructive example. The long thin landmass is centred on 42W and, at its broadest point, is no more than 750&nbsp;km from that meridian while the span in longitude reaches almost 50 degrees. Krüger–''n'' is accurate to within 1&nbsp;mm but the Redfearn version of the Krüger–''λ'' series has a maximum error of 1&nbsp;kilometre.

Karney's own 8th-order (in ''n'') series is accurate to 5&nbsp;nm within 3900&nbsp;km of the central meridian.