Non-Euclidean geometry / / H.S.M. Coxeter [[electronic resource]]

• Autore: Coxeter H. S. M (Harold Scott Macdonald), <1907-2003, >
• Pubblicazione: Washington : , : Mathematical Association of America, , 1998
• Edizione: Sixth edition.
• Descrizione fisica: 1 online resource (xviii, 336 pages) : digital, PDF file(s)
• Disciplina: 516.9
• Soggetto topico: Geometry, Non-Euclidean
• Note generali: Title from publisher's bibliographic system (viewed on 02 Oct 2015).
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: ""Front Cover""; ""NON-EUCLIDEAN GEOMETRY""; ""Copyright Page""; ""PREFACE TO THE SIXTH EDITION""; ""CONTENTS""; ""CHAPTER I. THE HISTORICAL DEVELOPMENT OF NON-EUCLIDEAN GEOMETRY""; ""1.1 Euclid""; ""1.2 Saccheri and Lambert""; ""1.3 Gauss, Wachter, Schweikart, Taurinus""; ""1.4 Lobatschewsky""; ""1.5 Bolyai""; ""1.6 Riemann""; ""1.7 Klein""; ""CHAPTER II. REAL PROJECTIVE GEOMETRY: FOUNDATIONS""; ""2.1 Definitions and axioms""; ""2.2 Models""; ""2.3 The principle of duality""; ""2.4 Harmonic sets""; ""2.5 Sense""; ""2.6 Triangular and tetrahedral regions""; ""2.7 Ordered correspondences"" ; ""2.8 One-dimensional projectivities""""2.9 Involutions""; ""CHAPTER III. REAL PROJECTIVE GEOMETRY: POLARITIES, CONICS AND QUADRICS""; ""3.1 Two-dimensional projectivities""; ""3.2 Polarities in the plane""; ""3.3 Conies""; ""3.4 Projectivities on a conic""; ""3.5 The fixed points of a collineation""; ""3.6 Cones and reguli""; ""3.7 Three-dimensional projectivities""; ""3.8 Polarities in space""; ""CHAPTER IV. HOMOGENEOUS COORDINATES""; ""4.1 The von Staudt-Hessenberg calculus of points""; ""4.2 One-dimensional projectivities""; ""4.3 Coordinates in one and two dimensions"" ; ""4.4 Collineations and coordinate transformations""""4.5 Polarities""; ""4.6 Coordinates in three dimensions""; ""4.7 Three-dimensional projectivities""; ""4.8 Line coordinates for the generators of a quadric""; ""4.9 Complex projective geometry""; ""CHAPTER V. ELLIPTIC GEOMETRY IN ONE DIMENSION""; ""5.1 Elliptic geometry in general""; ""5.2 Models""; ""5.3 Reflections and translations""; ""5.4 Congruence""; ""5.5 Continuous translation""; ""5.6 The length of a segment""; ""5.7 Distance in terms of cross ratio""; ""5.8 Alternative treatment using the complex line"" ; ""CHAPTER VI. ELLIPTIC GEOMETRY IN TWO DIMENSIONS""""6.1 Spherical and elliptic geometry""; ""6.2 Reflection""; ""6.3 Rotations and angles""; ""6.4 Congruence""; ""6.5 Circles""; ""6.6 Composition of rotations""; ""6.7 Formulae for distance and angle""; ""6.8 Rotations and quaternions""; ""6.9 Alternative treatment using the complex plane""; ""CHAPTER VII. ELLIPTIC GEOMETRY IN THREE DIMENSIONS""; ""7.1 Congruent transformations""; ""7.2 Clifford parallels""; ""7.3 The Stephanos-Cartan representation of rotations by points""; ""7.4 Right translations and left translations"" ; ""7.5 Right parallels and left parallels""""7.6 Study's representation of lines by pairs of points""; ""7.7 Clifford translations and quaternions""; ""7.8 Study's coordinates for a line""; ""7.9 Complex space""; ""CHAPTER VIII. DESCRIPTIVE GEOMETRY""; ""8.1 Klein's projective model for hyperbolic geometry""; ""8.2 Geometry in a convex region""; ""8.3 Veblen's axioms of order""; ""8.4 Order in a pencil""; ""8.5 The geometry of lines and planes through a fixed point""; ""8.6 Generalized bundles and pencils""; ""8.7 Ideal points and lines""; ""8.8 Verifying the projective axioms"" ; ""8.9 Parallelism""
• Sommario/riassunto: Throughout most of this book, non-Euclidean geometries in spaces of two or three dimensions are treated as specializations of real projective geometry in terms of a simple set of axioms concerning points, lines, planes, incidence, order and continuity, with no mention of the measurement of distances or angles. This synthetic development is followed by the introduction of homogeneous coordinates, beginning with Von Staudt's idea of regarding points as entities that can be added or multiplied. Tranformations that preserve incidence are called collineations. They lead in a natural way to isometries or 'congruent transformations'. Following a recommendation by Bertrand Russell, continuity is described in terms of order. Elliptic and hyperbolic geometries are derived from real projective geometry by specializing an elliptic or hyperbolic polarity which transforms points into lines (in two dimensions) or planes (in three dimensions) and vice versa.
• Titolo autorizzato: Non-Euclidean geometry
• ISBN: 1-61444-516-8
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910812285603321
• Lo trovi qui: Univ. Federico II
• Serie: MAA spectrum.