[Proj] Tmerc status with libproj4 and notes

[Paul B. Anderson]

- Hi All,

For those of you who have an interest, here is a small scale world map on 
the Gauss-Krüger projection:

http://www.csiss.org/map-projections/Cylindrical/Gauss_Kruger.pdf

I made this graphic using Philip M. Voxland's software he calls 'World' 
(MSDOS, Compiled Fortran).

- Paul B. Anderson

----- Original Message ----- 
From: "Gerald I. Evenden" <geraldi.evenden at gmail.com>
To: "PROJ.4 and general Projections Discussions" <proj at lists.maptools.org>
Sent: Monday, June 16, 2008 10:44 PM
Subject: [Proj] Tmerc status with libproj4 and notes


> Things on the libproj4-Transvers Mercator front are still progressing with
> another TM version finished today: ktmerc for Kruger.  Speed wise this 
> fits
> beteen etmerc and ftmerc and seems to give reasonable values out to 
> expected
> limits.  One problem did arise with ftmerc during testing that involved
> problems with the inverse operation.  This does not look like serious 
> fixing
> problem and a new release of libproj4 should be out later this week.
>
> Speaking of limits, it might be fun to re-explore the extremes of the
> Transverse Mercator since we had so much fun with the issue a couple of 
> years
> ago.  Since then we or at least I have still not seen any code that 
> delivers
> finite values for values of, say, 90W-0N.  All the versions in libproj4 
> still
> like to deliver stars for this kind of input.  Evidence seems permanently
> locked in the secret vaults and only available to the special few.
>
> For those who do not know what a global TM maps looks like draw a vertical
> line and place three equally space points on that line.  The points, from
> bottom to top are the south pole, north pole and the south pole again. Now
> draw two horizontal lines midway between the three poles: the equator. 
> Note
> that these two lines are part of the same thing yet never connect---kinda
> sounds like parallel lines that meet at infinity.  Now draw three more
> horizontal parallel lines through the three pole points.  THese lines are
> meridians at +-90 longitude.  Note that the right end of all five lines
> should meet at a point at 90E0N and similarly the left ends at a point 
> 90W0N.
>
> For the spherical case the situation of parallel lines meeting at infinity
> seems reasonable and there seems to be mathematical conditions for this
> state.  But for the elliptical case there is the currently publically
> unsubstatiated position that the end points are at a finite distance from 
> the
> central meridian of the map (the vertical line through the poles).
>
> This condition is analogous to finite polar extent cylindrical maps like
> Millers and flat pole pseudocylindricals.  Points at the poles are not 
> points
> but a line that borders the top and bottom of the map.  In the TM case a
> point at 90Ew0N is a line defining the left of right extent of the map.
> Regardless of whether there is a singularity at the edge plotting, 
> position
> becomes increasingly affected by the precision of the point's coordinates.
> It becomes apparent that even if it is possible to extend a TM map to the
> extreme edge, limiting the extent of the map in a manner similar to the
> standard Mercator is appropriate especially when considering distortion 
> along
> the left and right edges.
>
> Pursuing the issue of determining the real value of 90EW0N seems pointless 
> and
> only of interest to the diehard cartophile. Global extent TM maps are 
> easily
> and adequately handled by the spherical equations and thus leaves 
> elliptical
> usage only to largescale mapping and cadastal applications.
>
> -- 
> 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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