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The unit sphere $S$ in $\mathbb{R}^3$, given by the equation $x^2+y^2+z^2 = 1$, can be parametrized by $$(u,v)\mapsto (\cos u \cos v, \sin u \cos v, \sin v).$$ Under the above parametrization, points of $S \setminus \{(0,0,1),(0,0,-1)\}$ are in 1-to-1 correspondence with $(u,v) \in [-\pi,\pi)\times (-\pi/2,\pi/2)$.

Assuming the Earth is a perfect sphere, there should be a well-known map of the Earth (minus north and south poles) associated with this correspondence. Which one is it?

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That would be the equirectangular projection: https://en.wikipedia.org/wiki/Equirectangular_projection

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