My mother's master's thesis was on the projective plane! She took (plane) geometric theorems in the regular space, transformed to the projective plane, did another transform that is valid there (you can swap points and lines, or something like that - I don't recall exactly), then transformed back to non-projective space, and had a new theorem, which was true because the original theorem was true.

That’s cool! Probably the second transformation was to dualize. Points in the projective plane correspond to lines in the dual projective plane and vice versa.