Editing Transverse Mercator projection (section)

===Direct transformation formulae===
The direct formulae giving the Cartesian coordinates (''x'',''y'') follow immediately from the above. Setting ''x''&nbsp;=&nbsp;''y′'' and ''y''&nbsp;=&nbsp;−''x′'' (and restoring factors of ''k''{{sub|0}} to accommodate secant versions)
:<math>
\begin{align}
x(\lambda,\varphi)&= \frac{1}{2}k_0a
         \ln\left[
       \frac{1+\sin\lambda\cos\varphi}
        {1-\sin\lambda\cos\varphi}\right],\\[5px]
y(\lambda,\varphi)&= k_0 a\arctan\left[\sec\lambda\tan\varphi\right],
\end{align}
</math>
The above expressions are given in Lambert<ref name=lambert/> and also (without derivations) in Snyder,<ref name=snyder>{{cite book | author=Snyder, John P. | title=Map Projections—A Working Manual. U.S. Geological Survey Professional Paper 1395 | publisher =United States Government Printing Office, Washington, D.C. | year=1987}}This paper can be downloaded from [https://pubs.er.usgs.gov/pubs/pp/pp1395 USGS pages.] It gives full details of most projections, together with interesting introductory sections, but it does not derive any of the projections from first principles.</ref> Maling<ref name=maling>{{cite book | author=Maling, Derek Hylton | title=Coordinate Systems and Map Projections | publisher =Pergamon Press| year=1992|isbn=978-0-08-037233-4 |edition=second}}.</ref> and Osborne<ref name=merc>[https://web.archive.org/web/20130924093049/http://www.mercator99.webspace.virginmedia.com/ The Mercator Projections] Detailed derivations of all formulae quoted in this article</ref> (with full details).