[Proj] Relationship between transverse Mercator algorithms

[Mikael Rittri]

> >Relationship betweeen transverse Mercator algorithms by Charles Karney
> 
> I have never read such a clear and concise explanation of the differences 
> and similarities between the various TM algorithms.  Thank you!
> 
> [ --- ] 
>
> Irwin Scollar

Yes indeed! Thanks from me, too.

However, I'm wondering about a detail of how the errors are defined.

Charles Karney wrote (http://lists.maptools.org/pipermail/proj/2008-September/003737.html):
 
> For each set, define
>
>   dxn  = max(error in forward transformation,
>               discrepancy in forward and reverse transformations)
>           for nth order method (order e^(2*n))

Has the nominal error in the forward transformation been divided by the local 
scale factor?  

That is to say, I think understand the second part, the discrepancy 
between forward and inverse. If you start with P, do forward and then 
inverse, getting Q, then P and Q are both expressed in longitude and latitude, 
so the distance between P and Q is measured in true meters on the ground, 
I suppose.  

But if you do forward projection from P, getting (N0,E0) from your exact algorithm,
and (N1,E1) from a Krüger-like algorithm, then the error is - primarily - expressed
in projected coordinates.  Let us say there were a difference of 1 meter in projected
coordinates at mu = 60 degrees.  Then, since the local scale factor there would be 
about 2, the true difference on the ground is just half a meter.  In some sense. 

If you did not divide by the local scale factor, then I don't understand how you 
can take the max of the two kinds of error, since one is expressed in true meters 
and the other is expressed in nominal (projected) meters. 

I hope I am not talking nonsense here.  

Again, many thanks for the lucid explanations.

--
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