Plane Geometry - Bruce E. Shapiro

278 SECTION 48. PARALLEL LINES IN HYPERBOLIC GEOMETRYFigure 48.4: The critical number depends only on the distance of the pointfrom the line (theorem 48.5).Hence θ ∈ K ′ .By a similar argument, for any θ ∈ K ′ then θ ∈ K.Hence the K = K ′ , which means the critical number depends only ond(P, l).Since the critical number θ C depends only on the distance x = d(P, l) wecan write it as θ C = κ(x) for some function κ which we call the criticalfunction.Theorem 48.6 The critical functionis a decreasing function.κ(x) : (0, ∞) ↦→ (0, 90]Proof. Let a, b ∈ R such that 0 < a < b.Choose points P, A, B, D such that P A = a and θ = µ(∠AP D) is the angleof parallelism for P and −→ AB; and choose point A such that QA = b.We need to show that κ(b) ≤ κ(a) = θ.Choose E on the same side of ←→ AQ as D such that µ(∠AQE) = θ (see figure48.5).« CC BY-NC-ND 3.0. Revised: 18 Nov 2012