[Proj] Transverse Mercator algorithm
Oscar van Vlijmen
ovv at hetnet.nl
Tue May 27 07:18:44 EDT 2008
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
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