Does anyone know the proj4 Coordinte Reference System command line for Rectangular Polyconic Gores?

I am a GIS beginner using QGIS 1.8.0

I am trying to transform a world map from the standard WGS84 Coordinate Projection System to a set of rectangular polyconic gores, with a gore count of 12.

Please follow this link to see what a set of rectangular polyconic gores looks like:
http://www.mapthematics.com/ProjectionsList.php?Projection=130#rectangular

Rectangular polyconic gores do not seem to be on the standard menu of Coordinate Projection Systems in QGIS.

I have found the Custom Coordinate Reference System dialog box in QGIS. I need to type in a definition of ‘rectangular polyconic gores, with a gore count of 12’ in the proj4 format into the dialog box to make it work.

• Does anyone know exactly what it is I need to type?

The reason I am trying to make a world map in the ‘Rectangular Polyconic Gores’ (called RPG from now on) projection so that it can be cut out and pasted onto a sphere to make a world globe. There is a free Photoshop plugin called Flaming Pear available which performs this task, but if vector data such as coastlines are saved as a raster in QGIS and then imported into Photoshop and transformed from WGS84 to RPG, the lines will get progressively narrower towards the poles. Therefore I need to do the projection transformation in QGIS where the data is still in vector format. If QGIS is not a suitable tool, and I need to be using a different GIS program, please let me know. I know that Mapthematics’ Geocart program will perform this task, but it costs a lot of money.

As suggested in the QGIS manual, I have started reading ‘Cartographic Projection Procedures for the UNIX
Environment—A User’s Manual’ by Gerald I. Evenden (available here: pubs.usgs.gov/of/1990/0284/report.pdf )
and ‘Map projections – a working manual’by J.P. Snyder (available here: http://pubs.er.usgs.gov/publication/pp1395 ) in an effort to come up with the answer myself. However, I do not have proper mathematical or computer language training, so I find them virtually impossible to understand.

Below I have included links to all the related information I have found so far:

• Here is a well illustrated page with writing and equations,
  describing the RPG projection:
  http://members.shaw.ca/quadibloc/maps/mpo0502.htm

• I found this quote at http://www.maxneupert.de/luc/ It apparently
  describes the mathematical formula behind the RPG:

  ‘’Bugayavevskiy and Snyder reference a third Book, by Ginzburg and
  Salmanova from 1964, pages 171 ff, as the source of the following
  formula, quote:
  x = p sin δ, y = s + p(1 – cos δ)
  p = kR cot φ, δ = λ(sin φ)/k
  where k is a constant parameter, usually equal to 2 and s is the meridian distance of φ from the equator. If φ = 0, x =
  kRλ. To allow for the deformation from the curvature of the ball
  while maps are beeing pasted onto it, projection coordinates are
  multiplied by a constant coefficient determined from experience. End
  quote.’

• Here is an assembly language program describing RPG
  http://www.boehmwanderkarten.de/kartographie/netze/rta/proj_ern1_globussegmente.rta
  along with some explanatory photos:
  http://www.boehmwanderkarten.de/kartographie/is_netze_globussegmente.html
  These pages are both in German.
• Max Neupert also says that a mathematical description of the RPG
  Projection also turns up on:
  page 222 of Map Projections – A
  Reference Manual by Lev M. Bugayavevskiy and John P. Snyder, 1995
  ISBN 0-7484-0304-3
  and page 155 of Kartographische Netzentwürfe by
  Karlheinz Wagner, 1949 (and 1962) ASIN B0000BP33E However, both
  these books cost money.
• Here ias an illustrated article from 1909 , which includes, amongst
  other things, a discussion on how to create a set of RPGs
  http://www.genekeyes.com/CAHILL-1909/Cahill-1909.html

• Here is the proj4 command line for the ‘SAD69 Brazil Polyconic,
  ESPG:29101’ projection, which I cut and pasted from QGIS CRS dialog
  box. Apparantly this projection is a close relative of the RPG
  projection, even though they visually appear to be completely
  different:

  +proj=poly +lat_0=0 +lon_0=-54 +x_0=5000000 +y_0=10000000 +ellps=aust_SA +towgs84=-57,1,-41,0,0,0,0 +units=m +no_defs

If I find anything more Ill post it here. Id really love to get this one solved.

ogr2ogr -F GMT 50m_coastline.gmt ne_50m_coastline.shp
ogr2ogr -F GMT 50m_land.gmt ne_50m_land.shp

#!/bin/sh

outputfilename=globe.pdf
gores=12
landcolor=220/220/190
coastcolor=9/120/171

rm temp.ps

step=`echo 360/$gores | bc`
xshift=`echo "0.014*$step" | bc`

lon_right=`echo -180+$step | bc`
lon_center=`echo -180+$step/2 | bc`
GMT psxy -m -R-180/$lon_right/-90/90 -Jt$lon_center/0.014i -Bg10 -P 50m_land.gmt -K -G$landcolor --GRID_PEN_PRIMARY=0.05p,gray --FRAME_PEN=0.05p,gray -X"$xshift"i -K >> temp.ps
GMT psxy -R-180/$lon_right/-90/90 -Jt$lon_center/0.014i -m 50m_coastline.gmt -W0.2p,$coastcolor -O -K >> temp.ps
for i in `seq 2 $gores`
do  
    lon_left=`echo "-180+($i-1)*$step" | bc`
    lon_right=`echo "-180+$i*$step" | bc`
    lon_center=`echo "-180+($i-0.5)*$step" | bc`
    GMT psxy -m -R$lon_left/$lon_right/-90/90 -Jt$lon_center/0.014i -Bg10 -P 50m_land.gmt -K -G$landcolor --GRID_PEN_PRIMARY=0.05p,gray --FRAME_PEN=0.05p,gray -X"$xshift"i -O -K >> temp.ps
    GMT psxy -R$lon_left/$lon_right/-90/90 -Jt$lon_center/0.014i -m 50m_coastline.gmt -W0.2p,$coastcolor -O -K >> temp.ps
done
lon_left=`echo "-180+($gores-1)*$step" | bc`
lon_right=`echo "-180+$gores*$step" | bc`
lon_center=`echo "-180+($gores-0.5)*$step" | bc`
GMT psxy -m -R$lon_left/$lon_right/-90/90 -Jt$lon_center/0.014i -Bg10 -P 50m_land.gmt -K -G$landcolor --GRID_PEN_PRIMARY=0.05p,gray --FRAME_PEN=0.05p,gray -X"$xshift"i -O -K >> temp.ps
GMT psxy -R$lon_left/$lon_right/-90/90 -Jt$lon_center/0.014i -m 50m_coastline.gmt -W0.2p,$coastcolor -O >> temp.ps

ps2pdf temp.ps
pdfcrop temp.pdf $outputfilename