Isometries of the hyperbolic plane in the Beltrami–Klein disk model I am interested in the isometries of the hyperbolic plane in the Beltrami–Klein disk model. (https://en.wikipedia.org/wiki/Beltrami%E2%80%93Klein_model) The Wikipedia article does not say anything about the structure of the isometries in this model.
Since the isometries in the upper half-plane model are well-known, I did a change of variables to go from upper half-plane to the Beltrami–Klein disk. After some calculations, I determined that every isometry in the Beltrami-Klein disk model must be a projective transformation $\mathbb R  \mathbb P^2 \to \mathbb R  \mathbb P^2$ which maps the "unit disk" $\{ [x:y:1] \in \mathbb R \mathbb P^2 ~|~ x^2 + y^2 < 1 \}$ to itself. (The unit disk is, after all, the disk in the Beltrami-Klein disk model of hyperbolic space.)
Some questions I have:


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*Is every projective transformation of this form an isometry of the hyperbolic plane? (Is there an easy way to see this?)

*The set of such projective transformations is a subgroup of $SL(3,\mathbb R)$. Is there a nice characterization of this group?


Update: I just realized that every projective transformation which maps the unit disk to itself must be an element of $SO(2,1)$. Here is the reason: such a projective transformation must map the unit circle to itself, so the corresponding linear transformation $\mathbb R^3 \to \mathbb R^3$ must preserve the quadratic form $x^2+y^2-z^2$. Therefore, if the answer to Question 1 is "yes," then the answer to Question 2 should be $SO(2,1)$.

In addition to Lee Mosher's very nice answer, here are some texts that some of my friends have recommended:


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*John Ratcliffe: Foundations of Hyperbolic Manifolds 

*Cannon, Floyd,
Kenyon, Parry: Hyperbolic Geometry
