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Sky dome texture mapping help

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I wanted to use a sky dome in my simulation, but I don't have a sky dome mesh. I tried making a hemisphere, but I don't know how to generate texture coordinates for the vertices where y = 1 (in a DirectX coordinate system). Being a naive sort of fellow, I took the naive approach and generated a sky texture using the u texture coordinate to be the yaw and the v to be the pitch in spherical coordinate world. The entire hemisphere can be nicely unwrapped onto the skydome texture - except for the very top. I understand why this occurs. I feel like this is a problem that's been solved a long time ago and I just don't know where to look to find the solution.

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