[Proj] Transverse Mercator algorithm

[Mikael Rittri]

Charles Karney wrote:

> The Finnish algorithm is related.  The series relating [xi,eta] and [xi',eta'] are 
> just expansions (in eccentricity) of the transformation between two of the TM projections 
> described on p. 92 of Lee's 1976 monograph (namely what he calls the spheroidal 
> projection [xi',eta'] and the Gauss-Krueger projection [xi,eta]).

Am I right in thinking that the spheroidal projection [xi', eta'] is the same 
as Gauss-Schreiber?  Except that xi' and eta' are based on an ellipsoid whose 
radius is 1. (Sorry, I am not quite sure what I mean by ellipsoid radius in the 
previous sentence.)

Best regards,

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

-----Original Message-----
From: proj-bounces at lists.maptools.org [mailto:proj-bounces at lists.maptools.org] On Behalf Of Charles Karney
Sent: den 3 september 2008 23:00
To: geraldi.evenden at gmail.com
Cc: proj at lists.maptools.org
Subject: Re: [Proj] Transverse Mercator algorithm

 >> There is, however, one noteworthy difference.  The Swedish algorithm  >> is only approximately conformal, whereas the Finnish method is  >> conformal (to round-off).  This is easily remedied by substituting  >> the exact formula for the conformal latitude (as is done in the  >> Finnish report).
 >
 > In general, I am not that interested in answers to machine precision  > merely to about 0,0001mm or 0,001mm would be acceptable.

Fair enough.  Nevertheless, it does seem worth preserving the conformal property to machine precision -- especially when it can be done so easily.  (As an example of why this is useful, you can compute accurate transformations from one UTM zone to the neighboring one which rely on the fact that the transformation is conformal.  A 4-point fit gives 2um accuracy over a 10km x 10km area.)

 >> For my purposes, the exact method serves chiefly to gauge the  >> accuracy of the approximate method.  In particular, the error in the  >> Finnish and  >  > That is a useful and valuable asset.
 >
 >> The exact method and the derivations of the various series  >> approximations are all done in macsyma.  You'll probably need to have  >> macsyma installed if you want to use these.
 >
 > I have maxima and wxmaxima on my machine which seem to be related some  > time ago.  It is my understanding that macsyma is a commercial product  > and thus outside my pocketbook.  I do everything on the cheap.  ;-) It  > allows me to make everything I do free.

No need to pay.  I use the free SourceForge version.

It'll take a day or so to make my macsyma code for the exact transverse Mercator transformation presentable.  In the meantime, perhaps you can verify that your version of maxima functions the way I expect...

Download

     http://charles.karney.info/geographic/ellint.mac

Start maxima, and at the prompts, enter:

     load("ellint.mac")$
     ei(0.5b0+%i*0.3b0,0.1b0);

You should see something like:

     $ maxima
     Maxima 5.15.0 http://maxima.sourceforge.net
     Using Lisp CLISP 2.43 (2007-11-18)
     Distributed under the GNU Public License. See the file COPYING.
     Dedicated to the memory of William Schelter.
     The function bug_report() provides bug reporting information.
     (%i1) load("ellint.mac")$
     (%i2) ei(0.5b0+%i*0.3b0,0.1b0);
     (%o2) 2.96791299781263464924153153342948346508216221524596248948086\
     569795186664726317202946771827533813431304024592226608707871b-1 %i +
     4.99982389810326059482523070296819795538149014362475838573470325689\
     693122595700138979618859431972907386511841619290335085b-1
     (%i3)

If you get the same or equivalent output then you should be in good shape.

 > The math presented by Thompson and Lee seem quite at odds with several  > of the expanded range methods in that they express the problem in  > Jacobian functions.  Is there a parallel of have you looked into their  > methods?

The Finnish algorithm is related.  The series relating [xi,eta] and [xi',eta'] are just expansions (in eccentricity) of the transformation between two of the TM projections described on p. 92 of Lee's 1976 monograph (namely what he calls the spheroidal projection [xi',eta'] and the Gauss-Krueger projection [xi,eta]).

--
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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