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

[Oscar van Vlijmen]

>> <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.

My interpretation of "sphère de courbure" was:
the projection sphere has a radius equal to the radius of curvature at the
point of origin. But how is this sphere positioned? Tangent at the origin
(lat0, lon0) perhaps?
The "équatoriale" case has something to do with the projection sphere
positioned on the equator. In the "bitangente" case probably the projection
sphere touches the ellipsoid twice, but where?
Schreiber knew; he developed several double projections!

>> 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.
I understand that among others, ESRI has been busy developing such a TM. So
there must be a professional need for it.

> BTW: what did you use to get the "exact" values?
My own stuff.
I had a very hard time finding solid code. Several people are guarding the
principles as a secret and are deliberately vague. Or they are trying to get
solid money from it by selling software or books.
But I persevered in trying to get the Dozier show on the road.
You (mr. Evenden) already found one error in his code, but this error has to
be corrected in 3 places. It's the error of the elliptic parameter m, which
has to be the elliptic modulus k in 3 cases (tmfd, gk, tmid).
It appeared that the Dozier code - the complex Newton iteration - was
useless in some regions, especially large lon-lon0 and low latitudes. First
I used somewhat better elliptic functions from Cernlib. But, to more effect,
I concocted another iteration scheme based on TOMS algorithm 365, a very
slow downhill walkaround method, yet very powerful.
I checked my results with an on-line calculator from professor Schuhr, based
on the Klotz algorithms.
<http://gauss.fb1.fh-frankfurt.de/cgi-bin/cgi_gk>
This calculator fails for difficult areas (very large lon-lon0, small lat).
So I can only 'proof' my results in the difficult areas by doing a complete
round-trip and getting nearly the original data back.
And yes, I do the lon-lon0=90 degrees too.

The Dozier article:
<http://www2.bren.ucsb.edu/~dozier/publications.htm>