The underlying basic question, which I'm sure I'm not the first to ask, is what are the possible exotic/nonintuitive models of Euclid's axioms/postulates, outside the one where "lines" are interpreted as "geodesics". What seems to me to remain if you remove that metric aspect, is "lines and angles".
I'd like to know if there is a classification/list of "simplest" and "unexpected" interpretations of the terms "segment", "line", "point" and "angle" and such that:
- For any 2 points there is a segment joining them.
- Any segment can be continuously prolonged into a line.
- One can compare angles, and right angles are equal.
(About the fifth postulate, that's optional for me, and it's up to you if/in what form, to include it.)
My hope is obviously not "to get the final answer" to this question, I rather hope to get a list of not-well-known, mind-bending, models, possibly. Bonus points for cases where "lines" are very different from geodesics, or where the realization of the model is not on a metric space.
I guess that more or less alternatively, one could ask
- Is there a classification/theory of possible "angle structures" that can be added to spaces from incidence geometry? And if so, what are the nice properties of angle structures that have been popping up in such theories?