I don't know if it helps, but Photoshop has a filter called "Polar Coordinates" under the Distort panel.
It transforms this:
link
to this:
link
The problem seems to be that it doesn't really repeat the pattern, only stretches. As you can see, the middle resembles the original texture the most, but the closer you get to the edge the stretchier it appears.
Don't know how to approach the problem really... If I figure out something I'll let you know.EDIT: Okay, I figured out how to do it "perfectly".
Make a new document in Photoshop, match your canvas with the perimeter of your rotating sphere and double it once more (in this case 576 * 2 = 1152 * 2 =
2304)
Paste your texture in, flip it 180° and resize it to twice its size using the nearest-neighbor interpolation. Be sure to apply the transformation.
Then, transform it
AGAIN, this time to 50% its width and 200% its height. Fill out the remaining empty space with the texture, seamslessly.
Should look something like
this now.
Then apply the Polar Coordinates and add a full circle mask. Now resize the image's size to 50% and you're done!
Dunno if I explained that well enough but if you have questions be sure to ask.
This is the final product (resized by the forum, open in new window to see it in its full glory):
EDIT: Wait, no, it's still incorrect... But that was a mistake on my end. Instead of resizing the second time to 50% x 200%, make it 25% x 200%.