[Proj] Transverse Mercator algorithm

[Oscar van Vlijmen]

Mikael Rittri wrote:
> Hello,
> I am thinking about extending the accuracy of our Transverse Mercator. 
 > But is it worth the trouble?

In my opinion, there are 2 possibilities:

* High accuracy.
It appears extremely difficult to obtain any accuracy at all in some 
regions, especially around the poles and more than 80 degrees from the 
central meridian.
Then there is the difficulty of the bifurcation of the equator beyond 
some point (dependent on the eccentricity, but usually beyond 85 deg. 
or so from CM).
It costs a lot of calculation time and it is difficult to find a good 
working procedure. Several procedures work excellent to 60, 80 or even 
89 deg. from CM, but fail miserably beyond that.

* Speed.
It is entirely possible to find one or more procedures that give a 
reasonable speed, but only to about 60 deg. from the CM. If the latitude 
is larger than about 20 deg. then the distance to CM can be extended to 
let's say 80 deg. in some procedures.

I find the series expansion with hyperbolic functions a good compromise 
between speed and accuracy.
This procedure is followed by the geodetic services of Finland, Sweden 
and France. Publications with complete formulae (forward, inverse and 
even including meridian convergence and point scale factor) can be found 
on-line.
If I remember correctly Gerald Evenden has implemented one of these 
procedures as "ftmerc" in libproj - please ask him for details.

The Taylor series expansion as used by the current tmerc-type 
implementations leads to nowhere if you want to go farther than let's 
say 1 radian (57 deg) from CM. Many derivatives (probably to the 26th 
so) are needed, with extraordinary complexity and calculation times.

It would be interesting if the Engsager/Poder scheme were on-line or 
published through this list.

Oscar van Vlijmen