NationStates Jolt Archive

Here, I have a serious math problem.

I used to toy with an ultra-classical problem about spherical geometry and paths, and I found (autonomously, but I don't think I'm the first ever!) an infinite set of non-trivial solutions (infinite parallels on the sphere).

Now, my problem is: the set of solutions I found is a numerable family of sets, whose cardinality is the cardinality of the continuum. Hence, the cardinality of the whole class should be the continuum. The sphere, iirc, has the cardinality of the continuum, too. Hence it should be possible to build a bijection between the set of my solutions and the whole surface of the sphere (or, at the very least, the whole surface minus let's say the North Pole). I cannot figure out how to build it, though.

Any ideas?

http://www.statesymbolsusa.org/IMAGES/Virginia/cardinal_byOwnby1.jpg

... Hence it should be possible to build a bijection between the set of my solutions and the whole surface of the sphere (or, at the very least, the whole surface minus let's say the North Pole). I cannot figure out how to build it, though.

Any ideas?
Fourier transform? Express the sphere in terms of its representation in the fourier plane, then map the transformed lines to some particular frequency element, perhaps? (Totally out of the dark here, by the way)