<div> <pre style="font-size: 9pt;"><tt><tt>Gerald I. Evenden wrote:</tt></tt></pre></div>
<div> <pre style="font-size: 9pt;"><tt><tt>>The math presented by Thompson and Lee seem quite at odds with several of the <br>
>expanded range methods in that they express the problem in Jacobian <br>
>functions. Is there a parallel of have you looked into their methods?<br>
<br>
The Jacobian elliptic functions were a way of developing two intermediate projections, and they allow for a pithy expression of the mathematics. The last projection collapses into the elliptic integral development through a sequence of substitutions and cancellations.<br>
<br>
Regards,<br>
-- daan Strebe<br>
</tt></tt></pre></div>
<div> <br>
</div>
-----Original Message-----<br>
From: Gerald I. Evenden <geraldi.evenden@gmail.com><br>
To: Charles Karney <ckarney@sarnoff.com><br>
Cc: proj@lists.maptools.org<br>
Sent: Wed, 3 Sep 2008 11:51 am<br>
Subject: Re: [Proj] Transverse Mercator algorithm<br>
<br>
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<pre style="font-size: 9pt;"><tt>On Tuesday 02 September 2008 11:57:14 pm Charles Karney wrote:<br>
> Gerald I. Evenden wrote:<br>
> > At the moment I have three TMs which produce equivalent accuracy<br>
><br>
> except in<br>
><br>
> > the "problem area" where longitude approaches 90 degrees from the<br>
><br>
> central<br>
><br>
> > meridian on the equator. If you procedures produce finite values at<br>
> > lat=0,lon=90 I would very much like to see your efforts, especially<br>
><br>
> in terms<br>
><br>
> > of math development.<br>
> ><br>
> > You mention elliptic integrals but I did not see them in the pdf or<br>
><br>
> at least<br>
><br>
> > did not recognize them. Is the math complete in the pdf? It seems<br>
><br>
> like it<br>
><br>
> > is not. Of course the language is a barrier for me but the math is<br>
><br>
> easily<br>
><br>
> > understandable.<br>
> ><br>
> > A very quick scan of the pdf file makes me think the math is similar<br>
> > to what I call the Swedish version and I believe they made comment<br>
> > about their version being used in Finland (someone did ;-) ). But<br>
> > their method fails at 0,90.<br>
><br>
> Assuming that the "Swedish version" you refer to is<br>
><br>
><br>
> <a href="http://www.lantmateriet.se/upload/filer/kartor/geodesi_gps_och_detaljmatnin" target="_blank">http://www.lantmateriet.se/upload/filer/kartor/geodesi_gps_och_detaljmatnin</a><br>
>g/geodesi/Formelsamling/Gauss_Conformal_Projection.pdf<br>
<br>
I already have that pdf and yes, that is the one I refer to as the "Swedish <br>
TM".<br>
<br>
> then, indeed, the Finnish report I used is very similar. There is,<br>
> however, one noteworthy difference. The Swedish algorithm is only<br>
> approximately conformal, whereas the Finnish method is conformal (to<br>
> round-off). This is easily remedied by substituting the exact formula<br>
> for the conformal latitude (as is done in the Finnish report). I<br>
> implement the inverse of this transformation by Newton's method which<br>
> converges quickly to round-off.<br>
<br>
In general, I am not that interested in answers to machine precision merely to <br>
about 0,0001mm or 0,001mm would be acceptable. The only place in libproj4 <br>
where I went close to machine precision is the meridian distance function <br>
when I changed over to elliptic integrals in the computation and dropped the <br>
traditional sine power series expansion of the integral.<br>
<br>
Also, (not to start an argument) I am especially forgiving on precision of <br>
projection procedures covering large areas as in continental or global <br>
mapping. Precision is a property to be only concerned about in cadestral <br>
mapping and grid systems that traditionally cover small areas due to the <br>
increase of scale factor errors in the larger region (which has nothing to do <br>
with computational precision).<br>
<br>
> The Finnish method has only one essential approximation, namely,<br>
> substituting series expansions (accurate to e^8) for the transformations<br>
> between conformal and rectifying latitudes. As you point out, there is<br>
> no derivation for these series expansions in the report. However, I<br>
> have derived these expansions and extended them to e^16. The math here<br>
> is standard undergraduate or first year graduate level stuff and I used<br>
> macsyma to do the tedious parts of the algebra. In addition, I've<br>
> derived accurate expansions for the convergence and scale.<br>
><br>
> Any method based on such expansions will fail at lat=0, lon=90*(1-e)<br>
> because of a singularity in the complex plane in the transformation from<br>
> geodetic to conformal latitudes. (To be precise the inverse<br>
> transformation becomes multi-valued.) The singularity is mild but it<br>
> causes series expansions to fail. My slower "exact" method can navigate<br>
> around the singularity readily enough. In particular, there is no<br>
> singularity at 0N 90E. For what it's worth, the result here is<br>
><br>
> easting = 25953592.845413590 m<br>
> northing = 9997964.943020998 m<br>
> convergence = 90 deg<br>
> scale = 18.40462279198669<br>
<br>
The nature of this singularity blows libproj4's basic capabilty but I am still <br>
interested in this problem purely academically as I feel there is not a <br>
practical use for TM to the E-W limits---especially if you have to pay dearly <br>
in cpu time to get there.<br>
<br>
> WGS84: a = 6378137 m, f = 1/298.257223563, k0 = 0.9996, false easting =<br>
> false northing = 0 (i.e., 0N 0E -> 0,0).<br>
><br>
> For my purposes, the exact method serves chiefly to gauge the accuracy<br>
> of the approximate method. In particular, the error in the Finnish and<br>
<br>
That is a useful and valuable asset.<br>
<br>
> Swedish methods is 1mm if the distance to 0N 90E is 36 deg, 1m if the<br>
> distance is 20 deg, and 1km if the distance is 10 deg. I.e., they<br>
> "fail" well before 0N 90E is reached. Including additional terms in the<br>
> series will probably not change this appreciably (instead it will<br>
> increase the accuracy where it's already pretty good).<br>
><br>
> The exact method and the derivations of the various series<br>
> approximations are all done in macsyma. You'll probably need to have<br>
> macsyma installed if you want to use these.<br>
<br>
I have maxima and wxmaxima on my machine which seem to be related some time <br>
ago. It is my understanding that macsyma is a commercial product and thus <br>
outside my pocketbook. I do everything on the cheap. ;-) It allows me to <br>
make everthing I do free.<br>
<br>
The math presented by Thompson and Lee seem quite at odds with several of the <br>
expanded range methods in that they express the problem in Jacobian <br>
functions. Is there a parallel of have you looked into their methods?<br>
<br>
-- <br>
The whole religious complexion of the modern world is due<br>
to the absence from Jerusalem of a lunatic asylum.<br>
-- Havelock Ellis (1859-1939) British psychologist<br>
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</tt></pre>
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