# [Proj] Relationship between transverse Mercator algorithms

Mikael Rittri Mikael.Rittri at carmenta.com
Thu Sep 11 04:26:54 EDT 2008

```> >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
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mikael.rittri at carmenta.com
www.carmenta.com

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