Foundations Of Three-Dimensional Euclidean Geometry

Foundations Of Three-Dimensional Euclidean Geometry

by I. Vaisman
     
 

ISBN-10: 0824769015

ISBN-13: 9780824769017

Pub. Date: 08/01/1980

Publisher: Taylor & Francis

Foundations of Three-Dimensional Euclidean Geometry provides a modern axiomatic construction of three-dimensional geometry, in an accessible form. The method of this book is a graduated formulation of axioms, such that, by determining all the geometric spaces which satisfy the considered axioms, one may characterize the Euclidean space up to an isomorphism. A special…  See more details below

Overview

Foundations of Three-Dimensional Euclidean Geometry provides a modern axiomatic construction of three-dimensional geometry, in an accessible form. The method of this book is a graduated formulation of axioms, such that, by determining all the geometric spaces which satisfy the considered axioms, one may characterize the Euclidean space up to an isomorphism. A special feature of Foundations of Three-Dimensional Euclidean Geometry is the introduction of the parallel axiom at an early stage of the discussion, so that the reader can see what results may be obtained both with and without this important axiom. The many theorems, drawings, exercises, and problems richly enhance the presentation of material. Foundations of Three-Dimensional Euclidean Geometry is suitable as a textbook for a one- or two-semester course on geometry or foundations of geometry for undergraduate and beginning graduate students. Mathematics majors in M.A.T. programs will find that this exposition of a classical subject will contribute greatly to their ability to teach geometry at all levels; and logicians, philosophers, and engineers will benefit from this book's applications to their own interests.

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Product Details

ISBN-13:
9780824769017
Publisher:
Taylor & Francis
Publication date:
08/01/1980
Series:
Chapman & Hall/CRC Pure and Applied Mathematics Series, #56
Pages:
288
Product dimensions:
6.00(w) x 9.00(h) x 0.81(d)

Table of Contents

Chapter 0.Introduction1
1.The Axiomatic Method and Its Utilization in Euclidean Geometry1
a.The axiomatic method1
b.Axiomatics of Euclidean geometry6
2.Useful Notions from Other Mathematical Theories9
a.Binary relations9
b.Groups11
c.Fields and linear spaces13
d.Topological spaces17
Chapter 1.Affine Spaces21
1.Incidence Axioms and Their Consequences21
2.The Axiom of the Parallels and Its Influence on the Incidence Properties26
a.Affine spaces26
b.Projective spaces30
c.Desargues theorems32
3.The Fundamental Algebraic Structures of an Affine Space40
a.Vectors of an affine space40
b.The vector sum44
c.Scalars of an affine space47
d.Scalar algebraic operations51
e.Properties of the scalar field54
4.Coordinates in Affine Spaces59
a.The linear space structure59
b.Frames and coordinates62
c.Affine spaces over a field68
5.Affine Transformations73
a.Characteristic properties of affine transformations73
b.Special affine transformations76
c.The structure of the affine group80
d.Determination of affine transformations85
Problems90
Chapter 2.Ordered Spaces95
1.The Order Axioms and Their First Consequences95
a.Linear Order Properties95
b.Plane and Spatial Order Properties103
2.Polygons and Polyhedra110
a.Convex Polygons110
b.The Jordan separation theorem116
c.Problems on polyhedra122
3.Ordered Affine Spaces128
a.Equivalent order axioms128
b.Determination of the ordered affine spaces135
c.Orientation of ordered affine spaces143
4.Continuity Axioms150
a.The Dedekind continuity axiom150
b.Continuously ordered spaces152
c.The axioms of Archimedes and Cantor160
Problems167
Chapter 3.Euclidean Spaces171
1.The Congruence Axioms and Their Relations with the Incidence and Order Axioms171
a.A preliminary discussion of congruence171
b.Elementary properties of congruence of segments174
c.The Euclidean group of a line177
d.Plane congruence properties180
e.Miscellaneous congruence properties186
2.Euclidean Spaces193
a.Congruence and the parallel axiom193
b.The scalar field of an Euclidean space196
c.Euclidean structures and quadratic forms202
d.Real Euclidean space209
e.Absolute geometry211
3.A Short History of the Parallel Axiom219
a.Historical discussion219
b.Mathematical discussion225
4.The Independence of the Parallel Axiom239
Problems252
Hints for Solving the Problems257
References265
Index267

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