[Proj] Scale factor for Transverse Mercator

[Gerald I. Evenden]

On Tuesday 09 September 2008 3:01:01 am Mikael Rittri wrote:
> 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 ;-)

Mercator is probably the most abused projection.

If I recall correctly, Equador uses Mercator as a grid system projection and 
it would be an appropriate choice for any area on the equator with 
predominant E-W extent.

The tradition is to use Mercator for navigation charts presumably on the 
factor that you can chart rhumb line paths.  With modern navigation tools the 
choice is mute.

Mercator as a world map is an abominable choice.  Pick an equal area.

> 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?

Ellipsoidal form is proper for large scale navigation charts.

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

Does it make any difference.  Snyder liked to use an R in his spherical 
computation examples that was based upon a sphere with the same surface area 
of the Clarke 1866 ellipsoid.  For large scale maps try the mean radii of the 
parallel and meridian at the latitude of the map from your ellipsoid of 
choice.

> 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. --
> 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 _______________________________________________ Proj mailing
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-- 
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