Ryan, I sometimes wonder what could have become of you if someone had managed to harness that enquiring mind of yours at an early age.
As nearly as I can guess from your questions, you have discovered some of the basics of spherical geometry (and perhaps a bit of spherical trigonometry as well). Ordinary “Euclidian” geometry generally deals with spatial relationships on a flat surface or ordinary three-dimensional volumes. Spherical geometry works on very nearly the same principles, but on the surface of a sphere, like a football if you are an ant or a planet if you are human etc.
I cannot give you the whole run-down here and now (or any other time, not being a yachtsman!) but one of the first things more or less in geometry is to define a straight line; the shortest distance between two points. Any other line looks to us like a bend or curve. On a sphere that is not what looks like a straight line to us, but the line we get when we stretch a string on the surface. We get what we call a great circle, the shortest distance between points on that surface. This might sound nutty, but whenever we want to travel large distances fast and cheaply, we must travel along part of a great circle on Earth. If you have a globe map of the Earth and a piece of string, you can amaze yourself by finding the sort of route you follow from say, Dover to Lima, to Cape Town, or to Los Angeles. You go via some of the strangest detours!
As for your string of people who are NOT on a great circle; from the point of view of spherical geometry they are NOT on the shortest distance between points, but on a circular path. That might not seem logical, but try making the path still shorter, like a few metres from the pole, and suddenly you will see that it really is sort of circular, isn’t it?
Of course, there is no particular reason why the great circle has to be round the equator, ignoring the fact of course, that the planet isn’t a perfect sphere. And here we run into one of the most drastic differences between spherical and plane geometry: in plane geometry we can draw as many lines as we like parallel to any straight line ( though only one through any one point). On a sphere we cannot draw any parallel lines at all!!! Try it – any two different great circles will cross each other at two points at opposite side of the globe!
Get the picture?
I dunno. You might find it rewarding to look up "Spherical Geometry" in Wikipedia.
Cheers,
Jon
