This website was updated in August 2009
Introduction
The Transverse Mercator projection on the ellipsoid (TME) is the basis of many maps covering individual countries, such as Australia and Great Britain, as well as the set of Universal Transverse Mercator (UTM) projections covering the whole world (other than the polar regions). The (Redfearn) formulae relating latitude and longitude to grid coordinates for TME are given in many books and web pages but the full derivations are not readily accessible: the article on this web site remedies that deficiency.
Pedagogical reasons soon made it apparent that that it would be necessary to discuss the normal and transverse Mercator projection on the sphere before going on to discuss the normal and transverse Mercator projections on the ellipsoid. As a result the length of the article has doubled and redoubled, but I have resisted the temptation to cut out the details which would be straightforward for a professional but daunting for a `layman'. The resulting 200 page article is available as mercator.pdf: this is a large file of 2.3MB but a shorter article, which covers only the 'ordinary' Mercator projection, is also available as short.pdf.
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Motivation
My original motivation for writing the article was to discuss two common misconceptions about the maps based on the Transverse Mercator projections.
Most map users assume that the scale on the map is uniform. For example the Landranger maps of the Ordnance Survey of Great Britain (OSGB) are embellished with the words ``1:50000 scale, 2cm to 1km''. In fact the scale on the map varies slightly as a function of position and it is exact only on two slightly curved lines about 180km east and west of the central meridian at 2 degrees west. The variation from the nominal 1:50000 is very small but it is present nonetheless. This information is not presented on the maps themselves.
TME map projections are invariably covered by a set of grid lines and most users assume that they are aligned north-south. In general only one grid line runs true north-south, that associated with the central meridian of the projection. Thus in Britain only the grid line coincident with the meridian at 2 degrees west is true: all others deviate slightly from north-south meridians. Unlike the scale variation this distortion is described on the map sheets. Likewise the UTM series is a set of 60 projections, each covering a width of of 6 degrees in latitude, and the grid lines run true north-south only on the central meridians at 3, 9, 15, 21 ... degrees east.
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Sources
My sources for the TME formulae are to be found in a British journal---the Empire Survey Review dating from the nineteen forties to sixties. (The `Empire' was dropped from the title in the 1960s). The actual papers are fairly terse, as is normal for papers by professionals for their peers, and their perusal will certainly not add to the details presented here. Books on mathematical cartography are fairly thin on the ground, moreover they usually try to cover all types of projections whereas we are concerned only with Mercator projections. The few books that I found to be of assistance are listed in the bibliography.
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Mathematics
The mathematics involved is not difficult (depending on your point of view) but it does require the rudiments of complex analysis for one crucial step. On the other hand the algebra gets fairly heavy at times; Redfearn (see bibliography) talks of a ``a particularly tough spot of work'' and Hotine talks of reversing series by ``brute force and algebra''---so be warned. To make the article as self-contained as possible I have added a number of appendices covering the required mathematics.
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History
A brief historical note may be found here or in the pdf files.
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Chapters
Outlines of the chapters may be found here or in the pdf files.
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Appendices
This article is as self-contained as possible and to this end there are seven mathematical appendices. Some of these were developed for other uses so they are more general in nature.
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Latex source code
Anyone wishing to obtain the source code should please contact me at the email address given below. The code will be uploaded to this site when the errata have been reduced.
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To do
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Errata
Please email to address below. In each case give a page and line number. Perhaps someone will be enthusiastic enough to make a thorough check of everything!
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Queries
Please email queries to address below. I will make brief replies if possible.
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Contact
My email address is peter.1@mercator.myzen.co.uk
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