[Proj] New omerc approximations of Denmark System 34

[Mikael Rittri]

Hello,

I have tried to implement the two Denmark System 34 (S34) projections for Jutland and Zealand.
I managed to approximate Kristensen's rotated Transverse Mercator approximations [1]
by using +proj=omerc with alpha and gamma (new in Proj 4.8.0). Although I have made an
approximation of an approximation, I think the accuracy isn't worse, and possibly slightly
better for Jutland. The main advantage is that one can use Proj.4 directly, without Kristensen's
final rotation of the projected plane, which Proj.4 doesn't support.
     This may be a bit late, since System 34 is no longer recommended, but I post my
definitions here in case someone has a use for them:

System 34 for Jutland:

proj +proj=omerc +lonc=9.46 +lat_0=56.13333333 +x_0=-266906.229 +y_0=189617.957 +k=0.9999537 +alpha=-0.76324 +gamma=0.0 +datum=WGS84

System 34 for Zealand:

proj +proj=omerc +lonc=11.81 +lat_0=55.3333332 +x_0=-118947.024 +y_0=101112.545 +k=0.9999855 +alpha=1.190005 +gamma=0.0 +datum=WGS84

These will give an x-axis going eastward, which gives negative x-coordinates in Denmark,
and a y-axis going northward.  Traditionally, the positive x-axis points westward for these
systems, but I think the x-axis is usually flipped to go east in WMS, for example.

The official claim by KMS is that these coordinate systems are not properly defined
projections, and should be transformed via polynomials from ED50/UTM or ETRS89/UTM
coordinates. So, the polynomials will handle both the map projections and the datum shift [2].

However, in the beginning, the two projections seem to have been designed as some
kind of oblique Mercator projections, both with the central line azimuth near zero, and
with grid north along the oblique central line [3].
The advantage of the oblique central lines is that the grid lines of both systems are nearly
parallel. Indeed, they two coordinate systems nearly coincide, with a maximal difference of
about 60 m. The intention was that the coordinates should be similar to an earlier projection
that covered both Jutland and Zealand, at least similar enough that the same division into
map sheets could be used [4].

So omerc (Hotine oblique Mercator) from WGS84 works fine as an approximation, thanks to
the ability to specify both alpha and gamma separately. The worst errors I have seen, compared
to an official online converter [5], is 33 cm for Jutland and 21 cm for Zealand, but I haven't made
very many tests. However, my implementation for Jutland differs at most 30 cm from Kristensen's,
and my implementation for Zealand differs at most 6 cm from Kristensen's, so his accuracy
estimates should be roughly true for my implementations as well.
Kristensen writes [1]:
    Jutland: average error in x: 11 cm, in y: 14 cm, worst error 48 cm in Skagen.
    Zealand: average error in x: 5 cm, in y: 9 cm, worst error 18 cm.

Best regards,

Mikael Rittri
Carmenta
Sweden
http://www.carmenta.com

References:

[1] Leif Kahl Kristensen. Kp2000 - en nødvendighed eller ikke? Dansk Vejtidskrift 6/7 2001.
     http://asp.vejtid.dk/Artikler/2001/06-07%5C2930.pdf

[2] http://www.kms.dk/English/Geodesy+and+Surveying/Transformation/

[3] Knud Poder, Geodætisk Institut før og efter GIER.
     http://datamuseum.dk/site_dk/20110914/GI_GIER.pdf

[4] Leif Kahl Kristensen, Geoforum.
     http://kortdage.dk/Aktiviteter/Synspunkt/Synspunkt-blog.aspx?action=ShowArticle&ArticleID=9
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