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I'd like to make a world map using the Kravayskiy7, however QGIS seems to be unable to handle the projection, even though it is supported by proj4. 704900 (as is the EPSG for kav7) is not listed in the CRS and strangely when I try to define it manually using

+proj=kav7 +lat_0=0 +lon_0=0 +x_0=0 +y_0=0 +a=6371000 +b=6371000 +units=m

QGIS tells me that it's invalid (site for definition source). I always assumed that QGIS supported all projections that are featured by proj4. So why is QGIS behaving this way and how could I work around this?

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    With GDAL 2.1.0-dev I am also getting an error with gdalsrsinfo "+proj=kav7 +lat_0=0 +lon_0=0 +x_0=0 +y_0=0 +a=6371000 +b=6371000 +units=m". Obviously kav7 is not yet included in these builds. – user30184 Feb 15 '16 at 22:48
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The proj.4 definitions that are implemented in GDAL (which works in the background of QGIS) can be found here:

https://trac.osgeo.org/gdal/browser/trunk/gdal/ogr/ogr_srs_proj4.cpp

There is no code for kav7, hence the projection can not be supported.

You may ask on the GDAL developer mailing list for reasons. Maybe it is just too exotic so that noone implemented it. At least the inverse projection is defined, making the implementation easier than Winkel Tripel.

By the way, the Matplotlib Basemap Toolkit supports the kav7 projection: http://matplotlib.org/basemap/users/kav7.html, as well as the D3 geo projection package: https://github.com/d3/d3-geo-projection

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