I'm trying to find the average arc length between two random points on a unit sphere. The solution I've come up with is rather ugly. Consider a parametric surface:
$$X(u,v)=\sin u\cos v\\Y(u,v)=\cos u\cos v\\Z(u,v)=\sin v$$
(For a sphere, $u\in[0,2\pi]$, $v\in\left[\frac{-\pi}{2},\frac{\pi}{2}\right]$). Then, integrate the distance formula for two points $(x_1,y_1,z_1),(x_2,y_2,z_2)$along all three axes. This turns into a nasty integral, though.
$$F(\ldots)=\int_0^{2\pi} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} D(X_1,Y_1,Z_1,X_2,Y_2,Z_2)dv\mbox{ }du$$ $$\int_0^{2\pi}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}F(\ldots)\mbox{ }dv\mbox{ }du$$
(Where $D(\ldots)$ is the arc length formula.) I could plug this into mathematica or something, but I have a feeling this would
- take too long to compute
- be too complex
- probably also be wrong
What is a more efficient way to do this? How should I go about solving this problem?