[Proj] Insignificant difference between Swiss and Hungarian Oblique Mercator?

[Oscar van Vlijmen]

I didn't receive Mikael Rittri's message with the subject:
  [Proj] Insignificant difference between Swiss and
  Hungarian Oblique Mercator?
dated 2006-12-05
But I found it lurking in the archives. Of course, I could have accidentally
deleted the received message myself. In case others have missed it too: the
complete message is copied below.

As for the Hungarian EOV projection: I haven't seen any official test points
yet, so it is not clear if what I am doing is correct.
But the near-official formulae from the Department of Cartography and
Geoinformatics, Eötvös University, Budapest
<http://lazarus.elte.hu/gb/geodez/geodind.htm>
give for a
lat = 48; lon = 22;
an "x", "y" of:
299283.5538, 870209.6285 m

Monár Gábor & Timár Gábor give in an article to be found at:
<http://www.fomi.hu/internet/magyar/szaklap/2002/03/mar3.pdf>
an approximation for EOV with an oblique Mercator.
If I understand the parameters correctly:
ellipsoid: GRS67
lat0 = "47d 08m 39.8174s"; lonc = "19d 02m 54.8584s";
alfac = 90; gamma = 90;
k0 = 0.99993; x0 = -9370549.28432; y0 = 200000.00114;
set no_offset flag
omerc(lat,lon,lat0,lonc,alfac,gamma,k0,x0,y0);
Result: x = 870209.5387; y = 299283.5505 m
This differs by 9.0 cm and 3.3 mm with the correct values.
The authors state however that in the horizontal direction this
approximation is better than 0.17 mm.
Probably I didn't interpret the parameters correctly?

Another approximation with the Swiss omerc projection, where mr. Rittri
refers to, has never been discussed on the maptools list, but on the
r-sig-geo list.
<http://www.mail-archive.com/r-sig-geo@stat.math.ethz.ch/msg00600.html>
and <.../msg00601.html>
In my understanding the parameters are:
ellipsoid: GRS67
lat0 = 47.14439372222222; lon0 = 19.04857177777778;
x0 = 650000; y0 = 200000; k0 = 0.99993;
somerc(lat,lon,lat0,lon0,x0,y0,k0);
Result: x = 870209.6286; y = 299283.5528 m
Differences with the Hungarian values: 1.0 mm, 0.1 mm.

By the way, the PROJ.4 epsg init #23700 is WRONG, because k0 is not given.

By the road: somerc doesn't use an alpha as external parameter.


Possible conclusions:
* Don't use PROJ.4 epsg init #23700.
* The omerc approximation from Gábor is either too coarse or not easily
understandable.
* The somerc approximation as published seems very good, but probably not
within 0.014 mm as mr. Rittri seems to indicate.



ORIGINAL MESSAGE:

Mikael Rittri Mikael.Rittri at carmenta.se
Tue Dec 5 07:53:22 EST 2006

Hello,
I think I have implemented the Hungary EOV projection, which is a
generalization
of the Swiss Oblique Mercator (+proj=somerc).  In the Hungarian version,
you
can specify a normal parallel, distinct from the central one.  The
_normal_ 
parallel gives the center of the first step of the double projection
(ellipsoid -> sphere), while the _central_ parallel gives the center
of the second step (sphere -> oblique cylinder).

I've modified the code of the PROJ.4 somerc, to cover the Hungarian
usage as well, but I am rather surprised: the effect of having
normal parallel != central parallel is about 0.01 millimeters.

Yet, to quote Molnar and Timar,
http://www.fomi.hu/honlap/magyar/szaklap/2002/03/mar3.pdf ,
"its normal parallel slightly but intentionally differs ..." [from the
central one].

But I cannot understand the intention, if the difference in projected
coordinates 
is so small.  (And I don't read Hungarian; only the abstract is in
English).

I've read on this list that PROJ.4 users do use +proj=somerc for
Hungary,
but I thought the approximation would be worse.  Just to rule out the
possibility that I have made a silly mistake: could anyone confirm that
the approximation error is about 0.01 millimeters?

Best regards,
  --  Mikael Rittri

Hungary EOV specification: http://lazarus.elte.hu/gb/geodez/geodind.htm
(chapter 2 and 9).

My test points used the projected coordinates roughly in the four
corners of 
Hungary, which I unprojected with and without the distinct normal
parallel.
The differences were:
Easting Northing   Difference
400000, 0        -> 0.01350 mm
400000, 400000   -> 0.01399 mm
900000, 0        -> 0.01350 mm
900000, 400000   -> 0.01399 mm