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Map texture "manually" to a sphere

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I am writing a 2D game. Well. Mostly. All of the graphics are 2D, but I'd like to have a "ball" to roll around. Now, so far, I am content with SOL's solution in the SOL tutorials (projecting polar coordinates to the 2D surface). But I'd rather like to apply a texture to the ball.
So, the easy question is: How can I map a texture to a "circle" (well, a sphere, alright, but so far, I did not compute any normal vectors -- how could I achieve this? Do I need normal vectors for this?).

Where can I find a good example of sphere-texture-mapping? I googled, of course, but maybe I am looking for the wrong keys, because I _can_ find how to map a texture to "vertices" -- which, so far I don't have...

Am I talking gibberish again or can you actually follow my weird thoughts?! :)

Just ask for specifics, and I'll try to clarify; if you have a solution to my problem (or know where I have to look), I'd be thankful if you could help me!

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