Finding the leftmost, rightmost, top, and bottom, points, on a surface, of a sphere.

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So I'm making a 3D game, and the player is inside a glass sphere. I'm projecting a bunch of points onto the sphere, and I need to find the leftmost, rightmost, topmost, and bottommost points, so I can "draw" a square around them, to find the corners. I've tried to explain it using an image, but if it's not clear, I'm sorry. Please ask me to elaborate

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I hope you can help, and thank you for your time!