[Proj] Transverse Mercator algorithm

Charles Karney ckarney at sarnoff.com
Sat Sep 6 22:03:34 EDT 2008

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


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

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