[Proj] Scale factor for Transverse Mercator

Clifford J Mugnier cjmce at lsu.edu
Wed Sep 10 15:17:50 EDT 2008

The State Plane Coordinate Systems were originally conceived in the 1930s (from observed experiences in Indirect Artillery Fire Control during WWI), and were actually implemented in the 1940s-1950s.  The specification for Second-Order Transit Tape traverse survey closures at that time in the U.S. was 1:10,000.  In keeping with that criterion, the maximum scale factor at a point for a Zone was then intended to not be greater than that same amount.  Many Zones may exceed that by a smidgeon, though ...  The reason for that was with Surveyors having limited computational power at the time (pencil, paper, & tables of logarithms), a Surveyor could omit the correction for Scale Factor at a Point to convert from Grid Distances to Geodetic Distances and vice versa and still achieve survey closures to at least Third-Order accuracies, sometimes Second-Order.
The World Polyconic Grid used as the artillery grid system in the U.S. and the Netherlands East Indies for WWII by the U.S. Army was changed to the UTM in 1948 by Army Map Service.  The Scale Factor at a Point maximum of 1:2,500 at the Central Meridian was considered OK for Fire Control because for this application, the 6° Zones were limited in longitudinal extent because of the convergence angle between Grid North and Geodetic North - soldiers don't really care that much about the difference between Grid Distance and Geodetic Distance ... the blast radius of a 155mm Howitzer shell takes care of that little "problem."
The State Plane Coordinate Systems of the U.S.C.&G.S. and the Universal Grid Systems of the U.S. Army had entirely different criteria in their individual designs.  Both work just fine for their intended applications.  It has been observed by Military Geodesists that "A major battle will occur at every Grid Zone Boundary," and as a result they came up with the Zone-to-Zone Transformation Tables to extend a single UTM Zone up to 24° from the Central Meridian.
Cliff Mugnier


From: proj-bounces at lists.maptools.org on behalf of Gerald I. Evenden
Sent: Wed 10-Sep-08 12:17
To: Charles Karney
Cc: PROJ.4 and general Projections Discussions
Subject: Re: [Proj] Scale factor for Transverse Mercator

On Wednesday 10 September 2008 11:39:31 am Charles Karney wrote:
> Gerald I. Evenden <geraldi.evenden at gmail.com> wrote:
> > Let me pose the question: why are all of the state plane zones in the
> > US established to have very small scale errors (factors) over their
> > extent?
> Because an important application for state maps is defining land
> ownership, building roads, etc.
> > Why did DOD create the 6 degree zone if it were not to minimize the
> > scale error.
> DOD also wants bullets to go predicable distances in straight lines on
> maps.

Indeed, but in a sense this is also the same reason for the state plane
system---a mechanism to apply Cartesian mechanics to measurement problems.  I
must add that military usage is also dependent upon a wide level of mental
capability among the grunts on the ground and combined with the fog of war it
is a little easier to deal with these problems in Cartesian coordinates.  It
should probably be added that today's electronic warfare probably has a great
deal of effect on how maps are used in a modern combat---UTM is a relic of
mid 20th century warfare technology.

> However, both these applications are concerned with a subclass of human
> activities close to the surface of the earth.

To be sure.  But when we go to smaller scale we are more interested in
thematic mapping where we are less intersted in "measuring" in a
ruler-compass sense but visually understanding the "layout" of information. 
It is in this usage that I am much more in favor of equal area maps because I
feel that the sense of areal extent is much more important than conformality. 
The human eyeball does not really care if the graticule lines meet at right
angles everywhere.

> Google maps' use of Mercator is different class of application.  The
> requirements here are conformality and seamless scrolling over most of
> the globe with the normal application being navigating on roads rather
> than plotting great circle routes between continents.  Here the
> much-maligned Mercator seems to be an appropriate choice.

For what google is doing the projection makes little difference.  The roads
are probably in arc-node data sets where the distance along road segments
(arcs) are a matter of the data set and have nothing to do with the map
projection.  If they are using Mercator, they would probably adjust the
latitude of true scale as the latitude of the region displayed changes,  Thus
the "smooth" transition aspect is void.  If they didn't adjust lat_ts then a
street map of Brownsville, TX would look odd compared to a street map of
Fairbanks, AK or vice versa.

Another minor point with google, I bet they are *not* using a elliptical
projection software.  If they are, they are wasting a vast amount of computer

> I also read with interest Dozier's remark in NOAA Technical Report NESS
> 81 that the TM projection (extended beyond +/- 3deg) is useful for
> analyzing data from the NOAA A-G series of satellites.  I don't know why
> this might be beyond my ascertaining that these satellites are in polar
> orbits and TM can at least handle a single such orbit seamlessly.  So
> here is an application unanticipated (probably) by Gauss for the TM and
> beyond the narrow question of how to get a good map of Hanover.

There are lots of special projections in handling problems unique to
satellites.  Even in a polar orbit the path is not N-S on the map because the
earth turns as the satellite passes over head.

> I guess my point is that a library offering general-purpose projection
> capabilities should being agnostic on questions of what is a "good"
> projection.  Such issues are really in the application domain.

The question is not a moral question of good or bad but what is relevant. 
Sure, use TM beyond the 6 degree zone but don't mistreat it like Mercator and
make a world map out of it.  In addition, do not waste cpu clicks by using
the ellipsoid form when performing this extension.  If there is a moral issue
then it may be to *not* develop TM to 90/0 or to make the usage so costly
that it is never performed and thus minimize the carbon footprint of
cartographic 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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