Plane Geometry - Bruce E. Shapiro

Section 46Hyperbolic GeometryThe Hyperbolic Parallel Postulate (HPP) is mutually exclusive of the morefamiliar Euclidean parallel postulate. We restate them both here for reference.Axiom 46.1 (Hyperbolic Parallel Postulate) For every line l and everypoint P ∉ l there are at least two distinct lines m and n (m ≠ n) such thatP ∈ m and P ∈ n and m ‖ l and n ‖ l. (Note that m ̸‖ n because theyintersect at P !)Axiom 46.2 (Euclidean Parallel Postulate) For every line l and forevery point P that does not lie on l there is exactly one line m such thatp ∈ m and m ‖ l.One consequence of the failure of rectangles to exist in neutral geometry isthat the absence of Euclidean geometry implies the necessity of hyperbolicgeometry. For reference we recall the hyperbolic parallel postulate here. Inany model for neutral geometry, either the Euclidean parallel postulate orthe Hyperbolic parallel postulate will hold. This is a consequence of thefollowing theorem.Theorem 46.3 (Universal Hyperbolic Theorem (UHT)) If the EuclideanParallel Postulate fails, then the Hyperbolic Parallel Postulate istrue, i.e,¬(EP P ) ⇒ (HP P )The Universal Hyperbolic Theorem is sometimes stated differently: In anymodel of neutral geometry in which rectangles do not exist, the Hyperbolic263