a good description of the Cayley--Klein models especially about its homogeneity property

Question

Actuality, I'm working with conic in hyperbolic geometry and I'M looking for a good description of the Cayley--Klein models especially about its homogeneity property?

Answer 1

As the Cayley-Klein model of hyperbolic geometry, we set a Euclidean disc $\mathcal D$. Points in this model are explicate to be points interior to $\mathcal D$, and lines are explicate to be open chords of the bounding circle (i.e. chords without their endpoints).

The Cayley-Klein model is neither isometric nor conformal,however, the model does provide several distinct advantages over the Poincaré models.

First, it is promptly apparent that the hyperbolic axiom holds in this model. Given a chord $\ell$ of $\mathcal D$ and a point $P$ interior to the circle that is not on the chord, there are an infinitude of distinct chords passing through $P$ which do not intersect $\ell$.

Second, although the model is not conformal, constructing perpendicular lines in the cayley-Klein disk proves to be somewhat easier than in the Poincaré models.