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Average distance between two points on sphere!

  1. Jan 12, 2010 #1
    1. The problem statement, all variables and given/known data
    I had a work with average distance between points in circle, what was very funny... but now I must do a same thing for sphere:D this word "sphere" is so funny, maybe because of polish translation for this:D (something which reminds ball or sth)
    2. Relevant equations
    sphere parametr equations, defined integrals
    3. The attempt at a solution
    I describe sphere in parametr equations:
    [tex]\begin{cases}x=R\sin\phi\cos\varphi\\ y=R\sin\phi\sin\varphi\\ z=R\cos\phi\end{cases}[/tex]
    any two points [tex]A_1,A_2[/tex] on sphere can be described by [tex]\phi_1,\varphi_1,\phi_2,\varphi_2[/tex] parametrs, distance between points in all axis will be
    [tex]D\left(\phi_1,\varphi_1,\phi_2,\varphi_2\right)=\sqrt{\left(\sin\phi_1\cos\varphi_1-\sin\phi_2\cos\varphi_2\right)^2+\left(\sin\phi_1\sin\varphi_1-\sin\phi_2\sin\varphi_2\right)^2+\left(\cos\phi_1-\cos\phi_2\right)^2}[/tex]
    now I am going to simply this equation, do a quadruple integral of this function in [tex]\phi_1,\varphi_1,\phi_2,\varphi_2\in\langle0;2\pi\rangle[/tex] (this is [tex]\Omega[/tex]) and find average value, like this
    [tex]\frac{\iiiint\limits_\Omega f\left(\phi_1,\varphi_1,\phi_2,\varphi_2\right)\mbox{d}\Omega}{\iiiint\limits_\Omega\mbox{d}\Omega}[/tex]
    is it good idea, maybe I can do it easier? thank you!
     
  2. jcsd
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