I realised that using the script I had written to make the maps of Tau Ceti, it would be a simple matter to make an animation giving the illusion of a spinning planet. The orthographic azimuthal projection approximates this very well, and once one has access to the equirectangular map, there is nothing special about the two particular hemispheres used in the original maps — any meridian can be used as the centre meridian of the projection!
Edit: I have now added an animation of Gethen as well.
Edit 2: As an astute reader pointed out below, the original animations spun westward, meaning that the sun would rise in the west and set in the east, contrary to one’s expectation. There is no physical reason why the opposite should be preferred, but it feels more natural, and there is evidence in the books for this interpretation, wherefore I’ve updated the animations.
So, without further ado, here is Anarres (sorry about the long loading time!):
… and here is Urras:
Though the orthographic projection is the only one implemented that looks like a sphere, it is in no way special methodologically. As can be seen in the comparison in the original post, the stereographic projection is the furthest away from the orthographic, so I chose to make an animation of that as well, just for fun:
… and as a bonus I gave Gethen the same treatment:
The git repository has been updated with the example program.