Add +proj=mod_krovak projection method for Modified Krovak that applies to S-JTSK/05 in the Czech Republic

[rouault]

Validated with the test point given in EPSG Guidance Note 7-2

Also validated with CUZK online calculator at https://geoportal.cuzk.cz/(S(g4ipaut0ckzb5lellu153lvf))/Default.aspx?head_tab=sekce-01-gp&mode=TextMeta&text=wcts&menu=19

Unfortunately this calculator does not offer 'elementary' transformations, and cannot just validate Modified Krovak alone. The best we can do is to use ETRS89 <--> S-JTSK/05 / Krovak modified, which involves a Helmert transformation as well for the datum transformation.

• Validation of reverse modified Krovak:

  Online calculator:

  • Souřadnice/input: -5568990.91 -6050538.71 100
  • Source CRS: S-JTSK/05 + Bpv (-Y-XH /east-north)
  • Destination CRS: ETRS89 (BLh / DEG)
  • Výsledek/result: 50.208297081 16.848326835 143.762

  With PROJ:

  echo -5568990.91 -6050538.71 100 | bin/cs2cs -d 9 EPSG:5516 EPSG:4937 --3d
  50.208297081 16.848326833 143.143233569

Note the different on longitude is totally neglectable: 0.1 mm, as computed by:

  $ echo 50.208297081 16.848326835 50.208297081 16.848326833 | bin/geod -F "%.4f" -I +ellps=GRS80
  -90d 90d 0.0001

and that when outputing Krovak modified, the online calculator displays Y/X with
a precision of 1 cm.

The difference on Z is significant, but can be explained from the fact that
the online calculator using Baltic Height for S-JTSK/05 and thus apply a
Baltic Height<-->ETRS89 geoid that we don't apply here.

• Validation of forward Krovak using same point:

  Online calculator:

  • Souřadnice/input: 50.208297081 16.848326835 143.762
  • Source: ETRS89 (BLh / DEG)
  • Destination: S-JTSK/05 + Bpv (-Y-XH /east-north)
  • Výsledek/result: -5568990.91 -6050538.71 100

  With PROJ:

  $ echo 50.208297081 16.848326835 143.762 | bin/cs2cs -d 2 EPSG:4937 EPSG:5516 --3d
  -5568990.91 -6050538.71 100.62

[mwtoews]

There is an inconsistency between "Krovak Modified" and "Modified Krovak".

Guidance Note 7-2 mostly uses "Krovak Modified", with a few exceptions (e.g. "Modified Krovak East North")

An advantage of krovak_mod is that it nicely alpha-sorts after krovak.

Another more convoluted method is to keep only +proj=krovak with +modified option.

[rouault]

There is an inconsistency between "Krovak Modified" and "Modified Krovak".

yeah, I considered krovak_mod at first, but there was the risk of not noticing the _mod suffix, so I switched the mod_krovak to make things obvious. That said, if you feel that krovak_mod is preferable, I could switch to that

Another more convoluted method is to keep only +proj=krovak with +modified option.

I also considered that, but same reason, it is easy to miss it (and if people would copy&paste PROJ strings with +proj=krovak +modified to older PROJ releases, the +modified would be silently ignored, whereas +proj=mod_krovak will just error out). Furthermore, technically/formally, modified krovak takes extra parameters (the correction coefficients), which are currently hard-coded, because there's no point in requiring users to type them, given that the S-JTSK/05 is the single instance where this projection method is typed.

I was also wondering if we should set the default values of x_0 and y_0 to 5000000, which is what is used for S-JTSK/05, but was undecided, so I kept the 0 default value.

[mwtoews]

Re: "Krovak Modified" and "Modified Krovak". Looking further, there is no obvious "correct" form.

• In Czech it is "modifikovaném Křovákově", never the other way around
• English would normally prefer "Modified Krovak" too

So sure, let's stick with "Modified Krovak". See related suggested changes...