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Foundations of Three-Dimensional Euclidean Geometry : Chapman & Hall/CRC Pure and Applied Mathematics - Izu Vaisman

Foundations of Three-Dimensional Euclidean Geometry

Chapman & Hall/CRC Pure and Applied Mathematics


Published: 1st August 1980
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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. Book jacket.

Introductionp. 1
The Axiomatic Method and Its Utilization in Euclidean Geometryp. 1
The axiomatic methodp. 1
Axiomatics of Euclidean geometryp. 6
Useful Notions from Other Mathematical Theoriesp. 9
Binary relationsp. 9
Groupsp. 11
Fields and linear spacesp. 13
Topological spacesp. 17
Affine Spacesp. 21
Incidence Axioms and Their Consequencesp. 21
The Axiom of the Parallels and Its Influence on the Incidence Propertiesp. 26
Affine spacesp. 26
Projective spacesp. 30
Desargues theoremsp. 32
The Fundamental Algebraic Structures of an Affine Spacep. 40
Vectors of an affine spacep. 40
The vector sump. 44
Scalars of an affine spacep. 47
Scalar algebraic operationsp. 51
Properties of the scalar fieldp. 54
Coordinates in Affine Spacesp. 59
The linear space structurep. 59
Frames and coordinatesp. 62
Affine spaces over a fieldp. 68
Affine Transformationsp. 73
Characteristic properties of affine transformationsp. 73
Special affine transformationsp. 76
The structure of the affine groupp. 80
Determination of affine transformationsp. 85
Problemsp. 90
Ordered Spacesp. 95
The Order Axioms and Their First Consequencesp. 95
Linear Order Propertiesp. 95
Plane and Spatial Order Propertiesp. 103
Polygons and Polyhedrap. 110
Convex Polygonsp. 110
The Jordan separation theoremp. 116
Problems on polyhedrap. 122
Ordered Affine Spacesp. 128
Equivalent order axiomsp. 128
Determination of the ordered affine spacesp. 135
Orientation of ordered affine spacesp. 143
Continuity Axiomsp. 150
The Dedekind continuity axiomp. 150
Continuously ordered spacesp. 152
The axioms of Archimedes and Cantorp. 160
Problemsp. 167
Euclidean Spacesp. 171
The Congruence Axioms and Their Relations with the Incidence and Order Axiomsp. 171
A preliminary discussion of congruencep. 171
Elementary properties of congruence of segmentsp. 174
The Euclidean group of a linep. 177
Plane congruence propertiesp. 180
Miscellaneous congruence propertiesp. 186
Euclidean Spacesp. 193
Congruence and the parallel axiomp. 193
The scalar field of an Euclidean spacep. 196
Euclidean structures and quadratic formsp. 202
Real Euclidean spacep. 209
Absolute geometryp. 211
A Short History of the Parallel Axiomp. 219
Historical discussionp. 219
Mathematical discussionp. 225
The Independence of the Parallel Axiomp. 239
Problemsp. 252
Hints for Solving the Problemsp. 257
Referencesp. 265
Indexp. 267
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780824769017
ISBN-10: 0824769015
Series: Chapman & Hall/CRC Pure and Applied Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 268
Published: 1st August 1980
Country of Publication: US
Dimensions (cm): 23.37 x 15.55  x 2.31
Weight (kg): 0.59
Edition Number: 1