[Proj] Conversion between LCC_1SP and LCC_2SP

[Even Rouault]

Hi,

This is just for your information, in case others bang their heads as I did.
I wanted to write a dataset with a SRS expressed in Lambert Conformal Conic 1SP (LCC_1SP)
but the target file format (MapInfo .mif/.tab) only supports LCC_2SP apparently

The EPSG guidance( and Snyder ) mentions:
"""From the one standard parallel and its scale factor it is possible to
derive the equivalent two standard parallels and then treat the projection 
as  a  two  standard  parallel  Lambert  conical  conformal,  but this
procedure is seldom adopted.""""

Obviously, the proof was left as an exercice to the readers. So did I, as
I couldn't find existing implementations.

You can find the result here:
https://github.com/OSGeo/gdal/blob/e3f1db90d60968dd5a1b6618cfc4b1dfa1ea3521/gdal/ogr/ogrspatialreference.cpp#L7241

This requires an iterative method to solve StdParallel1 and StdParallel2.

So for example EPSG:27584
'+proj=lcc +lat_1=42.165 +lat_0=42.165 +lon_0=0 +k_0=0.99994471
+x_0=234.358 +y_0=4185861.369 +a=6378249.2 +b=6356515 +towgs84=-168,-60,320,0,0,0,0
+pm=paris +units=m +no_defs'

is equivalent to
'+proj=lcc +lat_1=42.76766346938247 +lat_2=41.56038784121728 +lat_0=42.165
+lon_0=0 +x_0=234.358 +y_0=4185861.369 +a=6378249.2 +b=6356515 +towgs84=-168,-60,320,0,0,0,0
+pm=paris +units=m +no_defs '

And the reverse (LCC_2SP -> LCC_1SP) also works ( requires modifying the false
northing in the general case ):
https://github.com/OSGeo/gdal/blob/e3f1db90d60968dd5a1b6618cfc4b1dfa1ea3521/gdal/ogr/ogrspatialreference.cpp#L7333

and EPSG:2154
'+proj=lcc +lat_1=49 +lat_2=44 +lat_0=46.5 +lon_0=3 +x_0=700000 +y_0=6600000
+ellps=GRS80 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs'

is equivalent to
''+proj=lcc +lat_1=46.5194302239868 +lat_0=46.5194302239868 +lon_0=3 +k_0=0.9990510286374693
+x_0=700000 +y_0=6602157.83881033 +ellps=GRS80 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs'

Even

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