## Curves on a Sphere – Epilogue

I just wanted to add a few more words about the benefits of the probabilistic solution. The nice thing is that it can be applied in all kinds of circumstances. For example, suppose that you have 2 curves with total length less then $2 \pi$. Then, they are either in one hemisphere together, or they are in two complementing hemispheres (or both).
Another thing is that we can let our curve be longer then $2\pi$, if we replace the great circle with something shorter, as long as the product of their length is less then $4 \pi^2$. To see this you only have to realize that the expectation we calculated is actually linear in the length of both curves. In other words, the formula should be $\mathbb{E}(X)= \ell_1 \ell_2 / 2 \pi^2$. As long as this is less then 2, we’re fine.

I guess that you can also use this trick on other surfaces, but I don’t have a good example (beyond Buffon’s needle again).