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).

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~ by Ori on November 5, 2008.

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