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Contents
Neutral and Non-Euclidean Geometries
David C. Royster
UNC Charlotte
Contents
List of Figures
List of Tables
The Origins of Geometry
Spherical Geometry
Logic and the Axiomatic Method
Introduction
Sets
Universal Sets and Compliments
Sentences and Statements
Sentence Connectives
Biconditionals and Combinations of Connectives
Quantifiers
Rules of Reasoning
Valid Arguments
Proof
Mathematical Systems
Proof
Proving Conditionals
Proving Biconditionals
Proving
Proof by Cases
Mathematical Induction
Proof by Contradiction
Proofs of Existence and Uniqueness
Proof Creativity
Euclid's Mathematical System
Incidence Geometry
Betweenness Axioms
Congruence Theorems
Axioms of Continuity
Neutral Geometry
Alternate Interior Angles
Weak Exterior Angle Theorem
Theorems of Continuity
Elementary Continuity Principle
Measure of Angles and Segments
Saccheri-Legendre Theorem
The Defect of a Triangle
The Work of Saccheri and Gauss
Saccheri
Gauss
Hyperbolic Geometry
The Hyperbolic Axiom and its Consequences
Angle Sums (again)
Similar Triangles
Classification of Parallels
Fan Angles
Limiting Parallel Rays
Hyperparallel Lines
Classification of Parallels
Proof of Claim and Asymptotic Triangles
Strange New Triangles
Inversion in Euclidean Circles
Models of Hyperbolic Geometry
Consistency of Hyperbolic Geometry
The Beltrami-Klein Model
The Poincaré Half-Plane Model
The Poincaré Disk Model
Isomorphism of Models
Constructions in the Poincaré Model
Return to the Klein Model
Area in Hyperbolic Geometry
Preliminaries
Requirements for an Area Function
The Uniqueness of Hyperbolic Area Theory
Angle of Parallelism
Hypercycles and Horocycles
The Pseudosphere
Hyperbolic Trigonometry
Circumference and Area of a Circle
Hyperbolic Analytic Geometry
More on Quadrilaterals
Coordinate Geometry in the Hyperbolic Plane
Index
About this document ...
droyster@math.uncc.edu