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Hilbert's axioms from Grundlagen der Geometrie involve notions of incidence, between-ness, segment congruence and angle congruence.

Consider the sub-theories of either Euclidean or hyperbolic geometry involving only the notions of incidence and between-ness. Of course, a priori, notions of congruence may occur in proofs (though not statements) of theorems solely about incidence and between-ness.

As a matter of fact, consider any 2-dimensional compact convex body $B$, with every boundary point extreme, say, but $B$ not affinely equivalent to a round disk. One can associate to $B$ an incidence geometry after the manner of Beltrami-Klein (with "lines" equal to interiors of chords). Such a geometry will not have an isomorphism to hyperbolic or Euclidean incidence geometries even though Hilbert's axioms about incidence and between-ness will hold! Indeed the incidence theory of such a quasi-Beltrami-Klein model actually determines $B$ up to an affine transformation. (Has this fact been recorded in the literature?)

Main Question: Can one formulate a finite, or at least an elegant set of incidence and between-ness axioms, extending Hilbert's, so as to capture the theory of Euclidean and/or hyperbolic incidence?

Added later: In light of Will Jagy's comment I'll sketch very briefly why I think the incidence geometry determines $B$ as much as it does, in case I'm obviously wrong. Given $B$, one can decorate it in many way with three families of evenly spaced parallel chords with triple intersections wherever intersections occur. The infinite of maximal configurations of this sort constitute approximations sufficient to recover $B$ up to affine.

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My first impression is that dropping angles leaves you no recourse, since you can take a homeomorphism of your $B$ to the B-K disk by taking a fixed internal point in $B,$ map it to $0,$ and map the rest of $B$ according to proportion of length along radii from the fixed central point to the same along radii out of $0.$ The lines in $B$ become the inverse images of segments in the B-K disk. – Will Jagy Sep 2 '12 at 4:52
Will: I don't follow your last sentence. Your homeomorphism will not take all straight chords to straight chords. – David Feldman Sep 2 '12 at 5:09
@David: You might want to check out the Wikipedia article on Ordered Geometry and the references that it mentions. This might get you started in the direction you want to go. – Robert Bryant Sep 2 '12 at 17:59
Thanks to one and all for the pointers. I should say I'm particularly interested in my quasi-B-K geometries and how they seem to know everything about convex bodies, and not just out to revisit the foundations of classical geometry. Of course I do want to understand how these ideas play out in the best understood cases. But hyperbolic geometry is probably more relevant than Euclidean. – David Feldman Sep 2 '12 at 20:49
@David: You should be aware that incidence and betweenness for the B-K geometry on a convex 2-dimensional body $B$ does not determine $B$ up to affine equivalence in general, but only up to projective equivalence. There are projectively equivalent convex bodies that are not affinely equivalent (of course, this cannot happen for ellipses, which are all affinely equivalent), and incidence and betweenness relations generated by chords will be the same for both bodies. Thus, for example, any two convex quadrilaterals are projectively equivalent, but generally, they won't be affinely equivalent. – Robert Bryant Sep 3 '12 at 18:40

2 Answers 2

Would you consider a positive answer for the Euclidean case being given by adding to the incidence and betweenness axioms the axioms for an affine plane with Desargues' configuration plus a completeness axiom along the lines of Dedekind's Axiom? These can be formulated purely in terms of incidence and betweenness and will characterize the affine plane based on the reals, I believe.

I don't know about the hyperbolic plane, but I suspect that you could probably define ideal and ultra-ideal points (adding incidence and betweenness axioms as necessary to make sure that these points are well-defined), add the axioms that would make the result be the Euclidean affine plane and then use some kind of polarity axiom to characterize the hyperbolic plane.

Have you consulted Greenberg's book? I don't have a copy handy, but I suspect that his references would lead you to a place where these questions are addressed.

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Maybe take Pappus instead Desargues, to ensure that the coordinates form a field instead of just a skew field. In any case, Pappus implies Desargues. – John Stillwell Sep 2 '12 at 15:59
@John: Oh, yes. That's a very good point! Indeed, I should have gone for Pappus. – Robert Bryant Sep 2 '12 at 17:38
@David, I thought about it overnight, and i am also not so sure that I know what you have in mind. I will leave the comment. I also agree with Robert that Marvin would know, but there is the problem of formulating the problem in terms he would like. I helped with a tiny part of the revisions for his fourth edition. Here is a page about his related article, a pdf can be downloaded from it:… – Will Jagy Sep 2 '12 at 18:15
So, if I understand, in terms of the coordinate ring, between-ness and Pappus will give us an ordered field. Then we also want it real-closed, so we can throw Dedekind completeness at it. But unfortunately, that assumption is second-order. I guess one doesn't expect finite first-order axioms for real-closed, but rather degree-by-degree axioms. Then these should translate into statements about configurations, perhaps in some clean and clever way? – David Feldman Sep 2 '12 at 20:44

The answer given to this question in the 20th century is quite complex, and a summary of what was done can be found in Victor Pambuccian, The axiomatics of ordered geometry I. Ordered incidence spaces, Expositiones Mathematicae 29 (2011) 24–66.

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