[Proj] Discovery: libproj4 stmerc = French Gauss-Laborde projection

[Gerald I. Evenden]

On Tuesday 13 June 2006 4:51 am, Oscar van Vlijmen wrote:
	 ...
> Going complex is not needed. Those few sums of complex sines can easily be
> computed doing Re and Im separately in reals.
>
> <http://www.ign.fr/telechargement/MPro/geodesie/CIRCE/NTG_73.pdf>
> NTG 73 describes 3 types of Gauss-Laborde projections. One of them, the
> "sphère de courbure", is equal to stmerc.

After looking more closely last night I began to come to that conclusion.  I 
am somewhat curious about the term "sphère de courbure" which an online 
translator gives: "curve of sphere."  Also, I am not yet quite sure what 
simplifies in the équatoriale and bitangente cases.

> If you throw a lot of lat, lon, lat0, k0 &c parameters to both and you get
> the same results everywhere, they must be the same. This is no rigorous
> proof of course.
>
> <http://www.ign.fr/telechargement/MPro/geodesie/CIRCE/NTG_76.pdf>
> NTG 76 describes algorithms for the transverse Mercator projection, using
> complex math. One of the advantages of this route is that you get very good
> results for large differences in lon-lon0.
> For instance, if you go 30 degrees away from the central meridian (lon0),
> at latitude 10 degrees, you'll find that tmerc departs around 60 m in x and
> around 30 m in y from the exact value.
> Routines like the French are only microns away from the exact values.
> Millimeter accuracy is still available to at least lon-lon0 = 70 degrees,
> and beyond that, only for smaller values of the latitude (say <30 deg) the
> function gets worse.

There is a solution out there that goes all the way but uses functions that 
are hard to find and/or need to be developed.  One fellow supposedly sped up 
the solution but I could not reproduce his results.

> At least three geodetic services use routines approximating the exact TM
> better than tmerc, DMA/NIMA/NGA and the like:
> 1) French IGN (see above)
> 2) Swedish Lantmäteriet
> <http://www.lantmateriet.se/upload/filer/kartor/geodesi_gps_och_detaljmatni
>n g/geodesi/Formelsaml%C3%ADng/Gauss_Conformal_Projection.pdf>
> 3) Finnish JHS
> <http://www.jhs-suositukset.fi/intermin/hankkeet/jhs/home.nsf/files/JHS154/
>$ file/JHS154.pdf>

I timed out trying to get to the above url.  I will try again later.

> Each follow a slightly different route, but the differences in the results
> are small.
>
> For small values of lon-lon0, say less than 6 degrees, tmerc et alii
> perform very well, so nothing can be gained using the above mentioned TMs.

One might question the practical need to go beyond 6 degrees.

BTW: what did you use to get the "exact" values?

> Let's give a testpoint:
> WGS84 ellipsoid
> lat=40; lon=70; lon0=0; lat0=0; x0=5e5; y0=0; k0=0.9996;
> x,y exact: 6289992.60347, 7531297.26735 m
> The French routines do: 6289992.60342, 7531297.26746
> The Finnish routines give: 6289992.60323, 7531297.26746
> By the way, the Finnish routines compute a meridian convergence of 58.289
> deg and a point scale factor of 1.43738.


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