Sphere/Dome question...

[sugarhouse]

Hopefully this hasn't been answered already, 'cause I couldn't find it in the forums...

I'm trying to discover some type of method/pattern/formula for applying flat, 2D graphics to a sphere.  (Actually, in my case it's 1/4 of a sphere, but I think the principle is the same.)  Typically, the graphics are on a rectangular sheet and if you place the sheet flat on the sphere, the ends of the rectangle will dip down, creating a "frownie face."  To remedy this, you need to cut darts along the top of the rectangle so you can straighten out the bottom and make it parallell to the "equator."

Unfortunately, every time I've done this, I've just made it up as I've gone along.  "Oh, sure - let's cut a dart about yay big right about there somewhere."  There has to be some way to calculate this process.  I thought about how they use gores to make globes.  (See attached.)  So if you were to cut apart a globe into that shape, the continents (or graphics in my case) would split apart as shown in the picture.

So I'd have to cut the graphics in the same shape as the top half of that picture, but I don't know how to calculate it.  I asked a co-worker about this, who replied, "You would think that Inventor would tell you how to do it for all the stinkin' money we paid for it."  I told him I'd look into it.

Any ideas?  (Let me know if this even makes sense or not.)  Thanks in advance for your time.