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.
Author(s): V. V. Prasolov, V. M. Tikhomirov
Publisher: American Mathematical Society
Year: 2001
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
