Axiomatic Projective Geometry

Bibliotheca Mathematica: A Series of Monographs on Pure and Applied Mathematics, Volume V: Axiomatic Projective Geometry, Second Edition focuses on the principles, operations, and theorems in axiomatic projective geometry, including set theory, incidence propositions, collineations, axioms, and coordinates.

The publication first elaborates on the axiomatic method, notions from set theory and algebra, analytic projective geometry, and incidence propositions and coordinates in the plane. Discussions focus on ternary fields attached to a given projective plane, homogeneous coordinates, ternary field and axiom system, projectivities between lines, Desargues' proposition, and collineations. The book takes a look at incidence propositions and coordinates in space. Topics include coordinates of a point, equation of a plane, geometry over a given division ring, trivial axioms and propositions, sixteen points proposition, and homogeneous coordinates. The text examines the fundamental proposition of projective geometry and order, including cyclic order of the projective line, order and coordinates, geometry over an ordered ternary field, cyclically ordered sets, and fundamental proposition.

The manuscript is a valuable source of data for mathematicians and researchers interested in axiomatic projective geometry.

Contenu

List of Symbols
Chapter I. Introduction.
§1.1. The Axiomatic Method
1.2. Notions from Set Theory
1.3. Notions from Algebra
1.4. Analytic Projective Geometry
1.5. Analytic Solid Projective Geometry
1.6. Vector Spaces Over a Division Ring
Chapter II. Incidence Propositions in the Plane.
§ 2.1. Trivial Axioms, Duality
2.2. Desargues' Proposition
2.3. Collineations
2.4. The First Quadrangle Proposition, Harmonic Pairs
2.5. Projectivities Between Lines
2.6. Pappos' Proposition
Chapter III. Coordinates in the Plane.
§3.1. Ternary Fields Attached to a Given Projective Plane
3.2. Ternary Field and Axiom System
3.3. Some Complementary Results
3.4. The Geometry Over a Given Ternary Field
3.5. Independence Results
3.6. Homogeneous Coordinates
Chapter IV. Incidence Propositions in Space.
§ 4.1. Trivial Axioms and Propositions
4.2. The Sixteen Points Proposition
Chapter V. Coordinates in Space.
§ 5.1. Coordinates of a Point
5.2. Equation of a Plane
5.3. Homogeneous Coordinates
5.4. The Geometry Over a Given Division Ring
Chapter VI. The Fundamental Proposition of Projective Geometry.
§ 6.1. The Fundamental Proposition
6.2. Summary of Results
Chapter VII. Order.
§ 7.1. Cyclically Ordered Sets
7.2. Cyclical Order of a Projective Line
7.3. Order and Coordinates
7.4. The Geometry Over an Ordered Ternary Field
7.5. A Counterexample
7.6. The Axiom of Archimedes
7.7. The Axiom of Continuity
Appendices
Index