Editing Transverse Mercator projection (section)

===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.