[Proj] Scale factor for Transverse Mercator

Gerald I. Evenden geraldi.evenden at gmail.com
Sun Sep 7 21:50:50 EDT 2008

To go back to another discussion related to extending the longitude range of 
TM I also wanted to emphasize the problem of the scale factor at increasing 
distance from the central meridian (CM).

For reference I put two graphs on the my proj4 website:




In both cases, the x axis is in terms of easting from the CM with degrees from 
the CM along the equator scale at the top.  The scale factor for TM is mainly 
a function of linear distance from the CM and northing or latitude  has 
little effect.

The 500er.png figure show scale factor in the range of principle usage of TM 
and demonstrates that the unadjusted percentage of scale error is less than 
0.15% over the standard 6 degree zones of most TM applications.  Note that in 
Scandinavian regions where the zones have been significantly increased in 
terms of longitude but the meaningful factor of easting has not increased.

In order to distribute the scale error more evenly though out the zone the 
scale multiplier k_0=0.9996 and now the scale error ranges between -0.05% to 

With larger scale errors arguments of using it for its conformal properties 
become rather meaningless and any projection with minimal visual distortion 
is usable.

ALSO, the spherical TM is quite useable at 40 degrees and gives nearly the 
same Cartesian values as well as nearly the same scale error.

When it gets to the bottom line, there seems to be little purpose in extending 
Transverse Mercator beyond its useful limits of 3 degrees.  And if there is a 
demand to ultra wide extent then the spherical form will server equally well.

To extend to the ultimate limit of 90 degrees is a bit of a problem only 
solvable with time consuming software and with results that are so distorted 
as to be unrecognizable.
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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