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Geometry A Metric Approach with Models

This book is intended as a first rigorous course in geometry. As the title indicates, we have adopted Birkhoff's metric approach (i.e., through use of real numbers) rather than Hilbert's synthetic approach to the subject. Throughout the text we illustrate the various axioms, definitions, a...

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Bibliographic Details
Main Authors: Millman, R.S., Parker, G.D. (Author)
Format: eBook
Language:English
Published: New York, NY Springer New York 1981, 1981
Edition:1st ed. 1981
Series:Undergraduate Texts in Mathematics
Subjects:
Geometry
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 6.5 Circles and Their Tangent Lines
  • 6.6 The Two Circle Theorem (optional)
  • 6.7 The Synthetic Approach (optional)
  • 7 The Theory of Parallels
  • 7.1 The Existence of Parallel Lines
  • 7.2 Saccheri Quadrilaterals
  • 7.3 The Critical Function
  • 8 Hyperbolic Geometry
  • 8.1 Asymptotic Rays and Triangles
  • 8.2 Angle Sum and the Defect of a Triangle
  • 8.3 The Distance Between Parallel Lines
  • 9 Euclidean Geometry
  • 9.1 Equivalent Forms of EPP
  • 9.2 Similarity Theory
  • 9.3 Some Classical Theorems of Euclidean Geometry
  • 10 Area
  • 10.1 The Area Function
  • 10.2 The Existence of Euclidean Area
  • 10.3 The Existence of Hyperbolic Area
  • 10.4 Bolyai’s Theorem
  • 11 The Theory of Isometries
  • 11.1 Collineations and Isometries
  • 11.2 The Klein and Poincaré Disk Models (optional)
  • 11.3 Reflections and the Mirror Axiom
  • 11.4 Pencils and Cycles
  • 11.5Double Reflections and Their Invariant Sets
  • 11.6 The Classification of Isometries
  • 11.7 The Isometry Group
  • 11.8 The SAS Axiom in ?
  • 11.9 The Isometry Groups of ? and ?
  • 1 Preliminary Notions
  • 1.1 Axioms and Models
  • 1.2 Sets and Equivalence Relations
  • 1.3 Functions
  • 2 Incidence and Metric Geometry
  • 2.1 Definition and Models of Incidence Geometry
  • 2.2 Metric Geometry
  • 2.3 Special Coordinate Systems
  • 3 Betweenness and Elementary Figures
  • 3.1 An Alternative Description of the Euclidean Plane
  • 3.2 Betweenness
  • 3.3 Line Segments and Rays
  • 3.4 Angles and Triangles
  • 4 Plane Separation
  • 4.1 The Plane Separation Axiom
  • 4.2 PSA for the Euclidean and Hyperbolic Planes
  • 4.3 Pasch Geometries
  • 4.4 Interiors and the Crossbar Theorem
  • 4.5 Convex Quadrilaterals
  • 5 Angle Measure
  • 5.1 The Measure of an Angle
  • 5.2 The Moulton Plane
  • 5.3 Perpendicularity and Angle Congruence
  • 5.4 Euclidean and Hyperbolic Angle Measure (optional)
  • 6 Neutral Geometry
  • 6.1 The Side-Angle-Side Axiom
  • 6.2 Basic Triangle Congruence Theorems
  • 6.3 The Exterior Angle Theorem and Its Consequences
  • 6.4 Right Triangles

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