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I am trying to average $r$ over the solid angle $\Omega$ in 3D. To start this I have expressed $r$ in terms of the angle $a$ and sides $x$ and $d$ in 2D with the help of the law of cosines: $r = x*cos(a) + d \sqrt{(1-x^2/d^2 Sin^2(a))}$ or $r = \sqrt{x^2 + d^2 - 2xd* Cos(a)} $. The next step that I tried to take was $\frac{1}{2\pi} \int_0^{2\pi} r da *\frac{1}{\pi}\int_0^{\pi}sin(\phi)d\phi$ to average over the solid angle $d\Omega = sin(\phi)d{\theta}d\phi$. This method gave me results in terms of elliptic integrals which is not what I was looking for. Is there a simpler (or completely different) method for doing averaging over the solid angle? Or should I reconsider my expressions for $r$? Thanks

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  • $\begingroup$ It is not clear what angle $\phi$ represents and why you are taking a space average for a planar figure. $\endgroup$ – Aretino Jan 13 at 19:26
  • $\begingroup$ With $\phi$ I mean the zenith angle, also the figure is supposed to be 3D but I tried expressing $r$ in 2D $\endgroup$ – Smitty Jan 13 at 19:41
  • $\begingroup$ Knowing the original 3D setting would help. You should also correct ${1\over\pi}$ to ${1\over2}$ in your integral formula, but the integral over $\phi$ could be simply discarded: you are averaging over angle $a$. $\endgroup$ – Aretino Jan 13 at 20:47
  • $\begingroup$ In addition, your second formula for $r$ is wrong. $\endgroup$ – Aretino Jan 13 at 20:59
  • $\begingroup$ Treating $a$ as an azimuthal angle cannot be correct, because $a$ can only vary between $0$ and $\pi$. Hence I suspect, even without knowing the setting in space, that you should simply regard $a$ as the zenith angle. $\endgroup$ – Aretino Jan 13 at 21:23

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