هندسه (ترجمه‌های مونوگراف‌های ریاضی)

This book provides a systematic introduction to various geometries, including Euclidean, affine, projective, spherical, and hyperbolic geometries. Also included is a chapter on infinite-dimensional generalizations of Euclidean and affine geometries. A uniform approach to different geometries, based on Klein's Erlangen Program is suggested, and similarities of various phenomena in all geometries are traced. An important notion of duality of geometric objects is highlighted throughout the book. The authors also include a detailed presentation of the theory of conics and quadrics, including the theory of conics for non-Euclidean geometries. The book contains many beautiful geometric facts and has plenty of problems, most of them with solutions, which nicely supplement the main text. With more than 150 figures illustrating the arguments, the book can be recommended as a textbook for undergraduate and graduate-level courses in geometry. Cover Selected Titles in This Series Geometry ISBN 0821820389 Contents Preface Introduction Chapter 1. The Euclidean World 1.1. The Euclidean line and plane Cartesian model of the Euclidean straight line and plane The Euclidean plane and complex numbers Some problems 1.2. n-dimensional Euclidean space The vector space Rn The affine space An and the Euclidean space En 1.3. Introduction to the multidimensional world of Euclidean geometry Affine varieties Determinants and volumes Simplices and balls Problems Chapter 2. The Affine World 2.1. The affine line and the affine plane Arithmetical model of the affine line Arithmetical model of the affine plane Linear equations on the plane Convex geometry on the plane and the theory of linear inequalities The fundamental theorem of affine geometry 2.2. Affine space. Theory of linear equations and inequalities 2.3. Introduction to finite-dimensional convex geometry The Carathéodory and Radon lemmas Helly’s theorem Problems Chapter 3. The Projective World 3.1. The projective line and the projective plane A model and some facts of the geometry of projective line The projective plane Pappus’ and Desargues’ theorems 3.2. Projective n-space Problems Chapter 4. Conics and Quadrics 4.1. Plane curves of the second order Metric, affine, and projective classification of second-order curves The ellipse, hyperbola, and parabola The ellipse, hyperbola, and parabola as conic sections 4.2. Additional remarks Fourth-degree equations The theorem about the conic passing through five points The theorem about the pencil of conics passing through four points The butterfly problem Hyperbolas with perpendicular asymptotes Pascal’s theorem Common chords of two conics inscribed in the same conic 4.3. Some properties of quadrics Two families of straight lines on a quadric Problems Chapter 5. The World of Non-Euclidean Geometries 5.1. The circle and the two-dimensional sphere: one- and two-dimensional Riemannian geometries The circle and the sphere Elementary spherical geometry Geometry of the n-sphere Riemannian, or elliptic, geometry 5.2. Lobachevsky geometry The Klein model of Lobachevsky geometry Linear-fractional transformations and stereographic projections Other models of Lobachevsky geometry Elementary hyperbolic geometry 5.3. Isometries in the three geometries Isometries of Euclidean space Isometries of the sphere Three types of proper motions of the Lobachevsky plane Problems Chapter 6. The Infinite-Dimensional World 6.1. Basic definitions 6.2. Statements of theorems 6.3. Proofs of the theorems 6.4. Concluding comments Addendum 1. Geometry and physics Projectiles move along parabolas (Galileo and Newton) The planets move along ellipses, and the asteroids, along second-order curves (Kepler and Newton) Geometry and special relativity (Einstein and Minkowski) 2. Polyhedra and polygons Convex polyhedra The Euler—Poincaré formula for the alternating sum of the numbers of faces (of various dimensions) of a convex polyhedron Dual polyhedra The Gram-—Sommerville formula for the alternating sum of solid angles at the faces of a convex polyhedron The Gauss—Bonnet theorem The Minkowski theorem The Cauchy theorem on rigid convex polyhedra Regular polyhedra The Cauchy formula The Steiner-Minkowski formula Polygons in Rm 3. Additional questions of projective geometry The complex projective space CPn The polar line of a point with respect to a curve in CP2 Projective duality Fixed points of projective transformations of the line and Steiner’s construction Projective involutions and harmonic quadruples of points and lines Problems 4. Special properties of conics and quadrics Confocal conics and quadrics Rational parametrizations of conics Poncelet’s theorem and the zigzag theorem The cross ratio of four points on a conic Problems 5. Additional topics of non-Euclidean geometries Paving the sphere, the plane, and the Lobachevsky plane by triangles Fundamental domains of the modular group Poincaré’s theorem about fundamental polygons The Lobachevsky space The quaternion model About the axiomatic approach to Euclidean and non-Euclidean geometries A brief excursion into the history of non-Euclidean geometry Conic sections in spherical and Lobachevsky geometries Parabolic mirrors in Lobachevsky geometry The volume of a simplex with vertices on the absolute Problems Solutions, Hints, and Answers Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Addendum Bibliography Author Index Subject Index Back Cover Presents an introduction to various geometries, including Euclidean, affine, projective, spherical, and hyperbolic geometries. This book also include a chapter on infinite-dimensional generalizations of Euclidean and affine geometries.