UID:
almafu_9960119291702883
Format:
1 online resource (xvii, 215 pages) :
,
digital, PDF file(s).
Edition:
1st ed.
ISBN:
1-316-04675-3
,
0-511-80620-5
Content:
This book gives a rigorous treatment of the fundamentals of plane geometry: Euclidean, spherical, elliptical and hyperbolic. The primary purpose is to acquaint the reader with the classical results of plane Euclidean and nonEuclidean geometry, congruence theorems, concurrence theorems, classification of isometries, angle addition and trigonometrical formulae. However, the book not only provides students with facts about and an understanding of the structure of the classical geometries, but also with an arsenal of computational techniques for geometrical investigations. The aim is to link classical and modern geometry to prepare students for further study and research in group theory, Lie groups, differential geometry, topology, and mathematical physics. The book is intended primarily for undergraduate mathematics students who have acquired the ability to formulate mathematical propositions precisely and to construct and understand mathematical arguments. Some familiarity with linear algebra and basic mathematical functions is assumed, though all the necessary background material is included in the appendices.
Note:
Title from publisher's bibliographic system (viewed on 05 Oct 2015).
,
Cover -- Half-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- Notation and special symbols -- Historical introduction -- Three approaches to the study of geometry -- An example from empirical geometry -- Nature of the book -- Plane Euclidean geometry -- The coordinate plane -- The vector space R^2 -- The inner-product space R^2 -- The Euclidean plane E^2 -- Lines -- Orthonormal pairs -- The equation of a line -- Perpendicular lines -- Parallel and intersecting lines -- Reflections -- Congruence and isometries -- Symmetry groups -- Translations -- Rotations -- Glide reflections -- Structure of the isometry group -- Fixed points and fixed lines of isometries -- Affine transformations in the Euclidean plane -- Affine transformations -- Fixed lines -- The affine group AF(2) -- Fundamental theorem of affine geometry -- Affine reflections -- Shears -- Dilatations -- Similarities -- Affine symmetries -- Rays and angles -- Rectilinear figures -- The centroid -- Symmetries of a segment -- Symmetries of an angle -- Barycentric coordinates -- Addition of angles -- Triangles -- Symmetries of a triangle -- Congruence of angles -- Congruence theorems for triangles -- Angle sums for triangles -- Finite groups of isometries of E^2 -- Introduction -- Cyclic and dihedral groups -- Conjugate subgroups -- Orbits and stabilizers -- Leonardo's theorem -- Regular polygons -- Similarity of regular polygons -- Symmetry of regular polygons -- Figures with no vertices -- Geometry on the sphere -- Introduction -- Preliminaries from E^3 -- The cross product -- Orthonormal bases -- Planes -- Incidence geometry of the sphere -- Distance and the triangle inequality -- Parametric representation of lines -- Perpendicular lines -- Motions of S^2 -- Orthogonal transformations of E^3 -- Euler's theorem -- Isometries -- Fixed points and fixed lines of isometries.
,
Further representation theorems -- Segments -- Rays, angles, and triangles -- Spherical trigonometry -- Rectilinear figures -- Congruence theorems -- Symmetries of a segment -- Right triangles -- Concurrence theorems -- Congruence theorems for triangles -- Finite rotation groups -- Finite groups of isometries of S^2 -- The projective plane P^2 -- Introduction -- Incidence properties of P^2 -- Homogeneous coordinates -- Two famous theorems -- Applications to E^2 -- Desargues' theorem in E^2 -- The projective group -- The fundamental theorem of projective geometry -- A survey of projective collineations -- Polarities -- Cross products -- Distance geometry on P^2 -- Distance and the triangle inequality -- Isometries -- Motions -- Elliptic geometry -- The hyperbolic plane -- Introduction -- Algebraic preliminaries -- Incidence geometry of H^2 -- Perpendicular lines -- Pencils -- Distance in H^2 -- Isometries of H^2 -- Reflections -- Motions -- Rotations -- H^2 as a subset of P^2 -- Parallel displacements -- Translations -- Glide reflections -- Products of more than three reflections -- Fixed points of isometries -- Fixed lines of isometries -- Segments, rays, angles, and triangles -- Addition of angles -- Triangles and hyperbolic trigonometry -- Asymptotic triangles -- Quadrilaterals -- Regular polygons -- Congruence theorems -- Classification of isometries of H^2 -- Circles, horocycles, and equidistant curves -- The axiomatic approach -- Sets and functions -- Groups -- Linear algebra -- Proof of Theorem 2.2 -- Trigonometric and hyperbolic functions -- References -- Index.
,
English
Additional Edition:
ISBN 0-521-27635-7
Additional Edition:
ISBN 0-521-25654-2
Language:
English
URL:
https://doi.org/10.1017/CBO9780511806209
