"Every geometry is a projective geometry." So is hyperbolic geometry a projective geometry?

Question

The great mathematician Arthur Cayley seems to have said "all geometry is projective geometry" (sorry no exact source, probably it is somewhere in Felix Klein's Erlangen program).

In projective geometry, non-intersecting lines do not exist (they all meet at infinity). In hyperbolic geometry on the other hand, through any point not on the line $\ell$, there are more than one line that does not intersect $\ell$, so at least some non-intersecting lines do exist. In all, they rather look incompatible together. A geometry cannot be both projective and hyperbolic. But still, it was Cayley who said it.

So how is hyperbolic geometry a projective geometry? I guess that it has to do with transformations/reflections and that ilk, but how does that relate to hyperbolic geometry in particular?

Answer 1

The so called Klein model for hyperbolic space gives a possible answer.

Consider an ellipse in the projective plane, or equivalently a quadratic form on the 3-dimensional vector space with signature (+,+,-).

The interior of the ellipse is a model for a hyperbolic plane where the lines are the intersections of usual projective lines with the ellipse. The distance between to point (A,B) can be expressed in terms of the cross ration of (A',A,B,B') where A',B' are the intersection point of E with the projective line through A and B.

The subgroup of the group of projective transformations which preserves this ellipse (this quadratic form) is exactly the group of isometry of the interior of the ellipse endowed with this hyperbolic distance.

The same model can be generalized in any dimension, with quadric instead of ellipse and quadratic form of signature (n,1).

In a few words $PSO(n,1) \subset PSL(n+1, \bf R)$ is the subgroup of projective transformations of the projective space of dimension $n$ which preserves the interior a quadric, this interior (endowed with usual lines) is a model of the hyperbolic space.

See https://en.wikipedia.org/wiki/Beltrami%E2%80%93Klein_model , and reference therein).

Answer 2

From the comments you included, I can see you are comparing the two geometries' synthetic axioms to see if one is a special case of the other. Of course, that is doomed to fail because the two lists contain mutually exclusive axioms about parallels (as you noticed.)

The real idea is that hyperbolic, affine and Euclidean geometries can be modeled as subsets of certain projective spaces founded upon vector spaces with suitable bilinear forms. The one for hyperbolic geometry is the Minkowski hyperboloid model, which realizes the hyperbolic plane upon the surface of a hyperboloid inside the projective plane.

Kaplansky's Linear algebra and geometry does a good job outlining all of this, although it is a little skimpy on details for a handful of topics along these lines.