Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?