This book presents plane geometry following Hilbert´s axiomatic system. It is inspired on R. Hartshorne´s fantastic book Geometry: Euclid and beyond, and can be considered as a careful exposition of most of chapter II of that book. It must be remarked that non-euclidean geometries, i.e. geometries not satisfying the fifth postulate of euclid, enter the scene, in a natural way, from the very beginning.
- Portadilla
- Title
- Copyright
- Dedication
- Preface
- Acknowledgments
- Contents
- 1 Introduction
- 1.1 A Short History of Geometry
- 1.2 What you will and will not learn in this book
- 1.3 Audience prerequisites and style of explanation
- 1.4 Book plan
- 1.5 How to study this book
- 2 Preliminaries
- 2.1 Proof methods
- 2.1.1 Methods for proving conditional statements
- 2.1.2 Methods for proving other types of statements
- 2.1.3 Symbolic representation
- 2.1.4 More examples of proofs
- 2.1.5 Exercises
- 2.2 Elementary theory of sets
- 2.2.1 Set operations
- 2.2.2 Relations
- 2.2.3 Equivalence relations
- 3 Incidence geometry
- 3.1 The notion of incidence geometry
- 3.2 Lines and collinearity
- 3.3 Examples of incidence geometries
- 3.3.1 Some basic examples of incidence geometries
- 3.3.2 The main incidence geometries
- 3.3.3 Generalizing the real cartesian plane
- 3.4 Parallelism
- 3.5 Behavior of parallelism in our examples
- 4 Betweenness
- 4.1 Betweenness structures, segments, triangles, and convexity
- 4.2 Separation of the plane by a line
- 4.3 Separation of a line by one of its points
- 4.4 Rays
- 4.5 Angles
- 4.6 Betweenness structure for the real cartesian plane
- 4.7 Betweenness structure for the hyperbolic plane
- 5 Congruence of segments
- 5.1 Congruence of segments structure and segment comparison
- 5.2 The usual congruence of segments structure for the real cartesian plane
- 5.3 The usual congruence of segments structure for the hyperbolic plane
- 6 Congruence of angles
- 6.1 Congruence of angles structure and angle comparison
- 6.2 Angle congruence in our main examples
- 6.2.1 Congruence of angles in the real cartesian plane
- 6.2.2 Congruence of angles in the hyperbolic plane
- 7 Hilbert planes
- 7.1 Circles
- 7.2 Book I of The Elements
- References
- Backcover
Text
Àudio
Adobe DRM
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