[Proj] Regarding projection stmerc and Gause-Laborde

[Gerald I. Evenden]

Some time ago someone claimed the stmerc and Gauss-Laborde were the same 
projection but at the time I the only material I had on the Gauss-Laborde was 
the NT/G 73 pdf.  The math in '73 was not easy and the tables for testing 
were obscure.  Given lazyness and other pressures I abandoned varifying the 
claim.

Recently Richard Didier offered his code for '73 for inclusion into libproj4 
so this resurrected the question about stmerc.  After doing a quick 
conversion of the code into libproj4 format I did some checking and again the 
messy tables of '73 confused things.  Didier pointed to the definition of the 
Reunion grid which I made into the following script:

# grid definition for L'lle de la Réunion from
# http://www.ign.fr/telechargement/MPro/geodesie/CIRCE/systemeReunion.pdf
RE="+ellps=intl +lon_0=55d32'E +lat_0=21d07'S +x_0=160000 +y_0=50000 +k_0=1.0"
lproj +proj=stmerc $RE -f %.4f <<EOF
55d30E 21dS
EOF

where the test longitude and latitude are from the the lefthand column of the 
table on page 3/3 of '73.  The x-y results are:

156534.1771     62916.9251

which depart only by .1mm in the y axis.

Soapbox time:

It is interesting to compare the computational layout of stmerc as compared to 
the documentation in NT/G 73.  Because this projection is a standard 
conformal double conversion of geographic coordinates of the ellipse to an 
equivalent sphere followed by a conformal projection of the sphere the 
forward code in stmerc looks like:

   lp = proj_gauss(lp, P->en); // standard geographic -> sphere conversion
    xy.x = P->aks0 * atanh(cos(lp.phi)*sin(lp.lam)); // standard sphere tmerc
    xy.y = P->aks0 * (atan2(tan(lp.phi),cos(lp.lam)) - P->chi);

I will not go further into my issues with the NT/G series.
-- 
The whole religious complexion of the modern world is due
to the absence from Jerusalem of a lunatic asylum.
-- Havelock Ellis (1859-1939)  British psychologist