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Given two spherical coordinates $\vec{r_1} = (1, \theta_1, \phi_2)$ and $\vec{r_2} = (1, \theta_2, \phi_2)$ the angle between them $\gamma$ can be calculated as $$\cos(\gamma)=\sin(\phi_1)\sin(\phi_2)\cos(\theta_1-\theta_2)+\cos(\phi_1)\cos(\phi_2)$$ If we then state that $\vec{r_2} = (1, \theta+d\theta, \phi+d\phi)$ we have $$\cos(\gamma) = \sin(\phi)\sin(\phi + d\phi)\cos(d\theta) + \cos(\phi)\cos(\phi+d\phi)$$ $$\cos(\gamma) \approx 1 - \frac{d\theta^2}{2}\left[\sin^2(\phi)+\sin(\phi)\cos(\phi)d\phi\right]$$
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Fri, 01 Nov 2013 14:55 GMT