Why are only greater circles considered lines in spherical geometry?

Question

I was looking for a simple explanation why only greater circles are considered as straight lines in spherical geometry (in the context of invalidating Euclid's fifth postulate) and not any circles of successively smaller radii that one can find on the surface of a sphere as one moves to poles. Thank you.

Answer 1

If you add circles on the surface that are not great circles, then you further degrade a basic condition that we want to be true (where possible) in a geometry:

Two distinct points should uniquely determine a line.

This is an axiom for ordinary geometries (euclidean, hyperbolic, projective), but of course it is not completely true for spherical, since antipodal points do not uniquely define a line. This is resolved if you move from spherical to projective by identifying antipodal points, as the two elements are no longer distinct.

But if you were to throw in these extra "lines" that you described, then every pair of distinct points has infinitely many lines through them. Furthermore, it will create extra lines when you identify antipodal points, and so the result (which used to be a projective geometry) is no longer a projective geometry because it fails the axiom about unique lines through two points.

Maybe this seems like a technicality, but just think about how often in regular geometry we feel free to draw a line through two points. If there isn't a unique line through two points, can you still draw one? Which one should you draw? You see, having too many lines introduces ambiguity.