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I am lacking the skill of visualizing the problem, picture here, to decide the right intervals. The way I do it currently is to try things but the technique fails with anything more complicated to 2D. So how do you know which angle correspond to which angle range?

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Let's break this into parts. Suppose there are two nodes, one $N$ in the north pole and one $S$ in the south pole. The distance $N-S$ is $\pi$. The distance $N-N$ is either $0$ or $2\pi$. Now, please, break this problem with nodes and graph-theoretically. See the paths are clearly different, the ending points vary: one is even and one is odd (trivially even if $N-N=0$ length accepted).

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    Should it be $\beta \leq \pi$ or are you trying to determine the area of some particular subset of the sphere?2011-05-17
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    To use your notation, $\alpha$ is longitude, and $\beta$ is co-latitude.2011-05-17
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    No what I mean is that by letting $\beta$ run to $\phi$ you are doing something odd... It's like when you're integrating the area of a unit square and integrating $\int_{0}^1 \int_{0}^x dx\,dy$. Now my hint to visualise this would be to draw a rectangle of size $\pi\times 2\pi$ and try to imagine how it folds around the sphere like a world map with coordinates $\phi,\theta$.2011-05-17
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    @Myself: sorry, typo, I meant $\pi$, not $\phi$.2011-05-17

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