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OverviewThis book examines the geometrical notion of orthogonality, and shows how to use it as the primitive concept on which to base a metric structure in affine geometry. The subject has a long history, and an extensive literature, but whatever novelty there may be in the study presented here comes from its focus on geometries hav ing lines that are self-orthogonal, or even singular (orthogonal to all lines). The most significant examples concern four-dimensional special-relativistic spacetime (Minkowskian geometry), and its var ious sub-geometries, and these will be prominent throughout. But the project is intended as an exercise in the foundations of geome try that does not presume a knowledge of physics, and so, in order to provide the appropriate intuitive background, an initial chapter has been included that gives a description of the different types of line (timelike, spacelike, lightlike) that occur in spacetime, and the physical meaning of the orthogonality relations that hold between them. The coordinatisation of affine spaces makes use of constructions from projective geometry, including standard results about the ma trix represent ability of certain projective transformations (involu tions, polarities). I have tried to make the work sufficiently self contained that it may be used as the basis for a course at the ad vanced undergraduate level, assuming only an elementary knowledge of linear and abstract algebra. Full Product DetailsAuthor: Robert GoldblattPublisher: Springer-Verlag New York Inc. Imprint: Springer-Verlag New York Inc. Edition: Softcover reprint of the original 1st ed. 1987 Dimensions: Width: 15.50cm , Height: 1.00cm , Length: 23.50cm Weight: 0.318kg ISBN: 9780387965192ISBN 10: 038796519 Pages: 194 Publication Date: 13 May 1987 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Active Availability: Out of stock  The supplier is temporarily out of stock of this item. It will be ordered for you on backorder and shipped when it becomes available. Table of Contents1 A Trip On Einstein's Train.- 2 Planes.- 2.1 Affine Planes and Fields.- 2.2 Metric Vector Spaces.- 2.3 Metric Planes.- 2.4 The Singular Plane.- 2.5 The Artinian Plane.- 2.6 Constants of Orthogonality.- 2.7 The Three Real Metric Planes.- 3 Projective Transformations.- 3.1 Projective Planes.- 3.2 Projectivities and Involutions.- 3.3 Matrix-Induced Projectivities.- 3.4 Projective Collineations.- 3.5 Correlations and Polarities.- 4 Threefolds.- 4.1 Affine Spaces.- 4.2 Metric Affine Spaces.- 4.3 Singular Threefolds.- 4.4 Nonsingular Threefolds.- 5 Fourfolds.- 5.1 Artinian Four-Space.- 5.2 Affine Fourfolds and Projective Three-Space.- 5.3 Nonsingular Fourfolds.- 5.4 The Three Real Fourfolds.- Appendix A Metageometry.ReviewsAuthor InformationTab Content 6Author Website:Countries AvailableAll regions |
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