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 ...