[Proj] Transverse Mercator algorithm

[Charles Karney]

Gerald I. Evenden wrote:
 > A couple of notes:
 >
 > 1. Of the etmerc, ktmerc and ftmerc extended range versions of TM,
 > etmerc appears to be the winner in precision as one approaches 90,0.
 > ...  Thus the combination of highest speed and best precision goes to
 > Denmark.

I tabulate here the errors in the JHS 154 algorithm for the Transverse
Mercator projection.  In addition I give

   * the errors when additional terms (up to O(e^16)) are included in the
     series to convert from conformal latitude to rectifying latitude.

   * the errors in the convergence and scale using the formulas given in

     http://lists.maptools.org/pipermail/proj/2008-September/003730.html

Errors are relative to an arbitrary precision formulation (see tm.mac).

The errors in libproj implementation of etmerc closely matches the
errors in 5th order method (dx5), except that there's a minimum
discrepancy of about 2um in the result given by the forward
transformation followed by the reverse.  I suspect that the reverse
transformation in etmerc is badly conditioned in some way.

Define mu = asin(sin(lam) * cos(phi))
           = angular distance from meridian

Errors are primarily a function of mu (or x).

For each set, define

    dxn  = max(error in forward transformation,
               discrepancy in forward and reverse transformations)
           for nth order method (order e^(2*n))

    dgam = max error in meridian convergence

    dk   = max relative error in scale

     set         dx4     dx5     dx6     dx7     dx8   dgam   dk
x<4e5, y<95e5 .21um     4nm     4nm     4nm     4nm   3e-8"  4e-7%%
x<5e5, y<96e5 .22um     4nm     4nm     4nm     4nm   9e-8"  1e-6%%
     mu<10     .35um     5nm     4nm     4nm     4nm   5e-6"  3e-5%%
     mu<15     .70um     6nm     4nm     4nm     4nm   4e-5"  2e-4%%
     mu<20     1.5um    11nm     4nm     4nm     4nm   2e-4"  5e-4%%
     mu<25     3.6um    25nm     4nm     4nm     4nm   7e-4"  2e-3%%
     mu<30     8.7um    71nm     4nm     4nm     4nm   3e-3"  3e-3%%
     mu<35      22um   .22um     6nm     4nm     4nm   7e-3"  6e-3%%
     mu<40      61um   .75um    11nm     4nm     4nm   0.02"  0.02%%
     mu<45     .18mm   2.8um    49nm     5nm     5nm   0.05"  0.03%%
     mu<50     .62mm    12um   .27um    10nm     6nm   0.2"   0.05%%
     mu<55     2.4mm    64um   1.8um    58nm     7nm   0.4"   0.1%%
     mu<60      12mm   .42mm    17um   .69um    34nm   1"     0.3%%
     mu<65      72mm   3.8mm   .22mm    13um   .83um   4"     0.6%%
     mu<70     .67m     56mm   5.1mm   .48mm    48um   18"    1.5%%
     mu<72     1.9m    .20m     22mm   2.6mm   .32mm   35"    2.4%%
     mu<74     6.6m    .88m    .12m     18mm   2.8mm   76"    3.9%%
     mu<76      27m    4.7m    .87m    .17m     35mm   3'     7.1%%
     mu<78     .14km    33m    8.4m    2.2m    .62m    8'     1.4%
     mu<80     1.0km   .36km   .13km    52m     21m    28'    3.3%
     mu<82      14km   8.1km   4.8km   3.0km   2.0km   2.4d   10%
     mu<84     390km   380km   390km   410km   440km   14d    15%

-- 
Charles Karney <ckarney at sarnoff.com>
Sarnoff Corporation, Princeton, NJ 08543-5300

URL: http://charles.karney.info
Tel: +1 609 734 2312
Fax: +1 609 734 2662