[Proj] Tmerc status with libproj4 and notes

[Gerald I. Evenden]

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