If the "Rectangular map" is in an Equidistant Cylindrical/Equirectangular projection, you can use G.Projector. Pick the "Sinusoidal" projection and select "Interrupted: 30° gores".
It's not identical to the projection you linked to (which appears to use transverse cylindrical gores rather than normal pseudocylindrical ones) but it's pretty close. If your base map is not in the right projection, or you really need that transverse output, things get complicated and a lot more work.



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