Remember that the rough draft of your second paper is due Tuesday, April 19.
I recommend reading W. F. Reynolds, Hyperbolic geometry on a hyperboloid, Amer. Math. Monthly 100 (1993) 442-455.
Read Chapter 7. Deemphasize "Inversion in circles", pp. 313-333. (But take note of how the obtained propositions are used to prove that the Poincare model satisfies the congruence axioms, pp. 318-321 and 327-330.)
To be more specific, I recommend:
Also, deemphasize "Projective nature ..." (pp. 333-346), but note Figure 7.35, which exhibits the isomorphism between the Poincare and Klein models in a very simple way.