[Proj] Extended range TM usage

Mikael Rittri Mikael.Rittri at carmenta.com
Mon Aug 25 11:00:28 EDT 2008

Dear Mr. Evenden,
Thanks for your efforts, but let me turn the question around:

What is the rational reason for using the Taylor series for 
ellipsoidal Transverse Mercator?  The Krüger formulas are 
almost as easy to implement, and seem better in all respects.  

Well-tested tradition?  But the Krüger formulas are from 1912, and 
have been recommended by the Swedish Land Survey at least since 1976. 

Speed? In my own implementation, I find no speed difference in
the inverse direction.  In the forward direction, I admit that 
Krüger's formulas are almost 3 times slower.  But no matter: 
if a point is within a region where a Taylor series gives at most 
1 millimeter error, then I use Taylor.  Elsewhere, I use Krüger.  
So my implementation is fast where it is safe to be so.  
(It is fairly simple to check which formulas to use.)
The safe area extends 6 degrees from the central meridan at the 
equator, but is wider (measured in longitude degrees) at higher 

But I'll try to give some answers to your question:

> My question is can anyone supply a rational reason for the practical 
> use of an elliptical TM projection with extended longitude range.  

1) How about conformal mapping of Norway?  You might say that the shape 
of Norway calls for an Oblique Mercator instead, but not if we include 
Svalbard, Jan Mayen (halfway between Iceland and Svalbard), Bovet Island 
(at 54° 26' S, 3° 24' E), and the Norwegian claims on Antarctica.  
A suitably placed ellipsoidal Transverse Mercator would be excellent 
where the population lives.  If the local scale factor is too bad on 
Jan Mayen, then the population there (if any) just have to learn to 
divide distances on the grid by the local scale factor.  
But if the round-trip accuracy fails on Jan Mayen (as I think it would
using the Taylor series), this might cause serious problems in a GIS system. 

2) Usage across a pole?  The longest direct flight in world goes between 
Singapore and New York.  Since these are on almost opposite meridians, 
a Transverse Mercator centered on Singapore would be a good choice to 
display possible flight routes.  The Taylor series cannot be used on 
the backside of the Earth.  But the Krüger formulas work there automatically
(in the Swedish description, the occurrences of atan must be replaced by atan2.)

3) GIS systems that are easier to use? Printed maps have a fixed scale, but 
a GIS system has a zoom button.  At some scale, it is usually necessary to 
switch to a suitable world projection, but this requires extra application 
code, which people often forget to insert.  If it is necessary to switch
from ellipsoidal Transverse Mercator to spherical Transverse Mercator at 
a certain scale, this is yet another thing people will forget.  (And no 
one reads documentation, by the way.) 

As they say, it is hard to make things fool-proof because fools are 
so ingenious.  That's why I like to have a single Transverse Mercator 
implementation that can be used in almost all scales. 
So, why not, if it has no real drawbacks?

Mikael Rittri
Carmenta AB
Box 11354
SE-404 28 Göteborg
Visitors: Sankt Eriksgatan 5
Tel: +46-31-775 57 37
Mob: +46-703-60 34 07 
mikael.rittri at carmenta.com
-----Original Message-----
From: proj-bounces at lists.maptools.org [mailto:proj-bounces at lists.maptools.org] On Behalf Of Gerald I. Evenden
Sent: den 25 augusti 2008 04:08
To: PROJ.4 and general Projections Discussions
Subject: [Proj] Extended range TM usage

I am finishing up the revised transverse Mercator section of libproj4's manual which includes three additional versions of the projection that give an extended longitude range for the projection.  These version persist in being singular at the equator and 90 degrees from the central meridian.  Hopefully, I may soon obtain a copy of Lee's work and include the Ultimate version.

The old Taylor series version (tmerc) of TM is shown to be quite adequate for cadestral applications as demonstrated by comparisons with the expanded versions,  That is, it has sub-millimeter accuracy within all longitude zones of standard applications that I am aware of.

At the current time I consider the work on the extended version academic and of little practical application for the following reason: scale error of TM beyond an easting of about 400km (approximately 3.5 degrees longitude at the
equator) would appear too large to make the projection useful in any application other than general mapping. If general mapping is the application then using the spherical form of tmerc should be adequate---especially for small scale maps.  The problem of the singularity of the projection at
(+-90,0) is not a problem because the distortion is so severe at the right and left edges the sides can be clipped.

My question is can anyone supply a rational reason for the practical use of an elliptical TM projection with extended longitude range.  The explanation should discuss how the projected coordinates are used and how the error factor is handled in application of the coordinates (like determining distance between points, azimuth, etc.).  Of course, how precise do you expect such measurements to be?

I would like to include some practical applications for using the added projections.


The whole religious complexion of the modern world is due to the absence from Jerusalem of a lunatic asylum.
-- Havelock Ellis (1859-1939) British psychologist _______________________________________________
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