[Proj] Transverse Mercator algorithm with goodaccuracy/speedtrade-off?

Mikael Rittri Mikael.Rittri at carmenta.com
Mon May 26 07:24:43 EDT 2008

Many thanks for all good advice.  This is quite a friendly 
and knowledgeable mailing list. 
I have looked at Dozier's paper now (thanks to Melita Kennedy 
for the link), and a summary of Poder-Engsager (thanks to 
daan Strebe). 
Actually, I had hoped that the Snyder/tmerc approach (or should 
I say Gauss approach) could be improved with just one or two more
extra terms in the series expansions.  But my understanding 
from Dozier's background text (and from Clifford Mugnier's mails
in the January 2003 archives), is now that it takes quite at lot 
of extra terms to get just slightly wider zones of accuracy. 
On the other hand, I find it hard to believe that Dozier's 
iterative algorithm is faster than the Krüger formulas from 1912
<http://www.lantmateriet.se/upload/filer/kartor/geodesi_gps_och_detaljmatning/geodesi/Formelsamling/Gauss_Conformal_Projection.pdf <http://www.lantmateriet.se/upload/filer/kartor/geodesi_gps_och_detaljmatning/geodesi/Formelsamling/Gauss_Conformal_Projection.pdf> >
As a simple comparison, I tried to count the number of 
transcendental function calls (TFCs) that will be made at 
run-time.  I am counting sin() and cos() of the same argument
as one TFC, and likewise for sinh() and cosh(). I have also assumed 
a few simple optimizations (for example, multiple-angle formulas 
can be used instead of evaluating sin(2t), sin(3t), sin(4t), ...).  
My results are (no warranties expressed or implied): 
Num. of TFCs  forw.  inverse 
tmerc           1      1+m   (m = number of iterations in pj_inv_mlfn, usually 1 or 2). 
Krüger          6       6
Poder-Engsager  7       ?    (see footnote)
Dozier         6+6n    6+4n  (n = number of iterations in cnewton, at least 1). 
So I would expect Krüger and Poder-Engsager to have about the 
same speed, and Dozier to be at least twice as slow. But of course, 
the ESRI people may have done clever optimizations of Dozier's 
For my own part, though, I think I may try to speed 
up my Krüger implementation a little more. 
Again, many thanks for all advice. 
Footnote: For Poder-Engsager (recommended by daan Strebe), 
I did not find the original source, only formulas for the 
forward direction given in 
http://www.fofodala.dk/projekter/9sem/2004_Flaadestyring_Himmerlands_Elforsyning.pdf <http://www.fofodala.dk/projekter/9sem/2004_Flaadestyring_Himmerlands_Elforsyning.pdf> , 
where credit is given to 
  Knud Poder and Karsten Engsager:
  Some Conformal Mappings and Transformations 
  for Geodesy and Topographic Cartography,
  KMS (Kort & Matrikelstyrelsen, Denmark) 1998. 

Mikael Rittri
Carmenta AB
Box 11354
SE-404 28 Göteborg
Visitors: Sankt Eriksgatan 5
Tel: +46-31-775 57 37
Mob: +46-703-60 34 07
mikael.rittri at carmenta.com

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