[Proj] Scale factor for Transverse Mercator

[Mikael Rittri]

Gerald I. Evenden wrote:

> Thus the combination of highest speed and best precision goes to Denmark.

Darn. And Sweden didn't do very well in the Olympic Games, either. 

> 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 serve equally well. 

I'm curious: what you say about scale factor variation would be equally true 
for ordinary Mercator - at least if I tilt my head 90 degrees like this ;-) 

So, would you say that an implementation of ordinary Mercator should 
use ellipsoidal formulas only within 3 degrees of latitude from the 
equator?  And that farther away, one should use spherical Mercator 
formulas?  

Speaking of spherical projection formulas, I'd like your opinion about 
what kind of geodetic datum that can be associated with a spherical
projection: 

1) An ellipsoid-shaped geodetic datum, or 
2) A spherical geodetic datum, which is not a real datum at all, but 
   we have to pretend so temporarily in order to compute the projection, or 
3) No datum at all, or 
4) ....

Best regards,

--
Mikael Rittri
Carmenta AB
Box 11354
SE-404 28 Göteborg
Visitors: Sankt Eriksgatan 5
SWEDEN
Tel: +46-31-775 57 37
Mob: +46-703-60 34 07 
mikael.rittri at carmenta.com
www.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 8 september 2008 03:51
To: PROJ.4 and general Projections Discussions
Subject: [Proj] Scale factor for Transverse Mercator

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:

http://members.verizon.net/~gerald.evenden/proj4/500er.png

and

http://members.verizon.net/~gerald.evenden/proj4/5000er.png

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 0.1%.

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.
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