# Average distance between two points on sphere!

1. Jan 12, 2010

### player1_1_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:
$$\begin{cases}x=R\sin\phi\cos\varphi\\ y=R\sin\phi\sin\varphi\\ z=R\cos\phi\end{cases}$$
any two points $$A_1,A_2$$ on sphere can be described by $$\phi_1,\varphi_1,\phi_2,\varphi_2$$ parametrs, distance between points in all axis will be
$$D\left(\phi_1,\varphi_1,\phi_2,\varphi_2\right)=\sqrt{\left(\sin\phi_1\cos\varphi_1-\sin\phi_2\cos\varphi_2\right)^2+\left(\sin\phi_1\sin\varphi_1-\sin\phi_2\sin\varphi_2\right)^2+\left(\cos\phi_1-\cos\phi_2\right)^2}$$
now I am going to simply this equation, do a quadruple integral of this function in $$\phi_1,\varphi_1,\phi_2,\varphi_2\in\langle0;2\pi\rangle$$ (this is $$\Omega$$) and find average value, like this
$$\frac{\iiiint\limits_\Omega f\left(\phi_1,\varphi_1,\phi_2,\varphi_2\right)\mbox{d}\Omega}{\iiiint\limits_\Omega\mbox{d}\Omega}$$
is it good idea, maybe I can do it easier? thank you!