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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?


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? – Myself May 17 '11 at 17:38
To use your notation, $\alpha$ is longitude, and $\beta$ is co-latitude. – J. M. May 17 '11 at 17:39
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 $\phi,\theta$. – Myself May 17 '11 at 17:45
@Myself: sorry, typo, I meant $\pi$, not $\phi$. – hhh May 17 '11 at 18:30

Look at a globe: To fix a position of a point on earth we give its geographical latitude $\theta$ and its geographical longitude $\phi$. The latitude $\theta$ runs from $90^\circ$ south (equivalent to $\theta=-{\pi\over2}$) at the south pole to $90^\circ$ north (equivalent to $\theta={\pi\over2}$) at the north pole. Along the equator the geographical latitude is $0^\circ$ or simply $0$. The geographical longitude $\phi$ is constant along meridians, it runs from $180^\circ$ west (equivalent to $\phi=-\pi$) to $180^\circ$ east (equivalent to $\phi=\pi$). The "dateline" $\phi=-\pi$ resp. $\phi=\pi$ is actually the same meridian on $S^2$, and the meridian $\phi=0$ is the meridian of Greenwich. We see that the sphere $S^2$ is covered essentially one-one by a $(\phi,\theta)$-coordinate system where $\phi$ runs from $-\pi$ to $\pi$ and $\theta$ from $-{\pi\over2}$ to ${\pi\over2}$. Note that under this representation the $\phi$-values of the north and south poles are undefined.

In many physical or technical situations, e.g. when studying the fine movement of a spinning top, it is more practical to have $\theta=0$ at the north pole. In this setup the angle $\theta$ takes values between $0$ and $\pi$. The formulas converting $(x,y,z)$-coordinates into spherical coordinates $(r,\phi,\theta)$ look a bit differently then, and the obvious symmetry $\theta\mapsto -\theta$ is not visible anymore.

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If you extend it to a globe, $d\theta$ represents the interval between two meridians of longitude, so should go from $0$ to $2\pi$. $d\phi$ represents the interval between two parallels of latitude, so should go over a range of $\pi$. On the globe, it is from $\frac{-\pi}{2}$ to $\frac{\pi}{2}$, but usually we measure from the pole and range over $0$ to $\pi$

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@hhh: Since we measure from the pole and not from the equator, in spherical coordinates, we speak of "co-latitude" and not latitude. – J. M. May 17 '11 at 17:42

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