Plane Geometry - Bruce E. Shapiro

SECTION 51. THE POINCARE DISK MODEL 303Proof. Let P and Q be the ends of the diameter containing A such thatQ ∗ A ∗ O ∗ P . Thend = ln(AP )(OQ) AP= ln(AQ)(OP ) AQe d = APAQ = 1 + AO1 − AOe d (1 − AO) = 1 + AOe d − (AO)e d = 1 + AOe d − 1 = (AO)(e d + 1)Lemma 51.6 (Euclidean Geometry) Let AB be a chord of a circle Γ, andlet C be the intersection of the tangent lines to Γ at A and B. Then △ABCis isoceles.Theorem 51.7 (Boylai Lobachevsky Formula) The angle of parallelismθ(d) in the Poincare Disk model of Hyperbolic geometry satisfiestan θ(d)2= e −dProof. Since θ(d) depends only on the distance we are free to choose any lineand any point. Let l be a diameter and let P be a point on a perpendiculardiameter. We will computer θ(d) = θ(OP ), as in fig. 51.4.Figure 51.4: Proof of the Boylai-Lobachevsky Formula. The edge of thePoincare Disk is the dotted line.The limiting parallel ray through P will be an arc of circle that is tangentto the diameter at of the Poincare disk at C. From lemma 51.6 △P CR isRevised: 18 Nov 2012 « CC BY-NC-ND 3.0.