A. Ebgorelov GEOMETRY z y Mir Publisl1ers·Moscow ABOUT THE BOOK This is a manual for the stu dents of universities and teach ers' training colleges. Contain ing the compulsory course of geometry, its particular impact is on elementary topics. The book is, therefore, aimed at pro fessional training of the school or university teacher-to-be. The first part, analytic geometry, is easy to assimilate, and actually reduced to acquiring skills in aJ!plyiug algebraic methods to elementary geometry. The second part, differential geometry, contains the basics of the theory of curves and sur faces. Thfl third part, foundations of geometry, is original. The fourth part is devoted to certain topics of elementary geometry. The book as a whole must in terest the reader in school or university teacher's profession. Albgorelov GEOMETRY GEOMETRY A. B. IloropeJioB fEOMETPHH Ha~aTeJibCTBO «HayRa•> MocRBa A. Pogorelov GEOMETRY MIR PUBLISHERS MOSCOW Translated from the Russian by Leonid Levant, Aleksandr Repyev and Oleg Efimov First published 1987 Revised from the 1984 Russian edition Ha aHeAuucnoM nablne Printed tn the Union of Soviet Socialist Republics © IIBAaTeJibCTBO «HayKa&. rnaBHa.JI pep;aKn;II.JI cpiiBRKo MaTeMaTRqecKOii: JIIITepaTypbl, 1984 © English translation, Mir Publishers, 1987 CONTENTS 10 Preface Part One. Analytic Geometry 11 Chapter I. Rectangular Cartesian Coordinates in the Plane 11 1. Introducing Coordinates in the Plane 11 2. Distance Between Two Points 12 3. Dividing a Line Segment in a Given Ratio 13 4. Equation of a Curve. Equation of a Circle 15 5. Parametric Equations of a Curve 17 6. Points of Intersection of Curves 19 7. Relative Position of Two Circles 20 Exercises to Chapter I 21 Chapter n. Vectors in the Plane 26 1. Translation 26 2. Modulus and the Direction of a Vector 28 3. Components of a Vector 30 4. Addition of Vectors 30 5. Multiplication of a Vector by a Number 31 6. Collinear Vectors 32 7. Resolution of a Vector into Two Non-Collinear Vectors 33 8. Scalar Product 34 Exercises to Chapter II 36 Chapter Ill. Straight Line in the Plane 38 f. Equation of a Straight Line. General Form 38 2. Position of a Straight Line Relative to a Coordinate System 40 3. Parallelism and Perpendicularity Condition for Straight Lines 41 4. Equation of a Pencil of Straight Lines 42 5. Normal Form of the Equation of a Straight Line 43 6. Transformation of Coordinates 44 7. Motions in the Plane 47 8. Inversion 47 Exercises to Chapter III 49 Chapter IV. Conic Sections 53 f. Polar Coordinates 53 2. Conic Sections 54 3. Equations of Conic Sections in Polar Coordinates 56 6 Contents 4. Canonical Equations of Conic Sections in Rectangu- lar Cartesian Coordinates 57 5. Types of Conic Sections 59 6. Tangent Line to a Conic Section 62 7. Focal Properties of Conic Sections 65 8. Diameters of a Conic Section 67 9. Curves of the Second Degree 69 Exercises to Chapter IV 71 Chapter V. Rectangular Cartesian Coordinates and Vectors in Space 76 1. Cartesian Coordinates in Space. Introduction 76 2. Translation in Space 78 3. Vectors in Space 79 4. Decomposition of a Vector into Three Non-coplanar Vectors 80 5. Vector Product of Vectors 81 6. Scalar Triple Product of Vectors 83 7. Affine Cartesian Coordinates, 84 8. Transfonnation of Coordinates 85 9. Equations of a Surface and a Curve in Space 87 Exercises to Chapter V 89 Chapter VI. Plane and a Straight Line in Space 95 1. Equation of a Plane 95 2. Position of a Plane Relative to a Coordinate System 96 3. Normal Form of Equations of the Plane 97 4. Parallelism and Perpendicularity of Planes 98 5. Equations of a Straight Line 99 6. Relative Position of a-Straight Line and a Plane, of Two Straight Lines 100 7. Basic Problems en Straight Lines and Planes 102 Exercises to Chapter VI 103 Chapter VII. Quadric Surfaces 109 1. Special System of Coordinates 109 2. Classification of Quadric Surfaces 112 3. Ellipsoid 113 4. Hyperboloids 115 5. Paraboloids 116 6. Cone and Cylinders 118 7. Rectilinear Generators on Quadric Surfaces 119 8. Diameters and Diametral Planes of a Quadric Surface 120 9. Axes of Symmetry for a Curve. Planes of Symmetry for a Surface 122 Exercises to Chapter VII 123 Part Two. Differential Geometry 126 Chapter VIII. Tangent and Osculating Planes of Curve 126 1. Concept of Curve 126 2. Regular Curve 127 3. Singular Points of a Curve 128 4. Vector Function of Scalar Argument 129 Contents 7 5. Tangent to a Curve 131 6. Equations of Tangents for Various Methods of Specifying a Curve 132 7. Osculating Plane of a Curve 134 8. Envelope of a Family of Plane Curves 136 Exercises to Chapter VIII 137 Chapter IX. Curvature and Torsion of Curve 140 1. Length of a Curve 140 2. Natural Parametrization of a Curve 142 3. Curvature 142 4. Torsion of a Curve 145 5. Frenet Formulas 147 6. Evolute and Evolvent of a Plane Curve 148 Exercises to Chapter IX 149 Chapter X. Tangent Plane and Osculating Paral:oloid of Surface 151 1. Concept of Surface 151 2. Regular Surfaces 152 3. Tangent Plane to a Surface 153 4. Equation of a Tangent Plane 155 5. Osculating Paraboloid of a Surface 156 6. Classification of Surface Points 158 Exercises to Chapter X 159 Chapter XI. Surface Curvature 161 1. Surface Linear Element 161 2. Area of a Surface 162 3. Normal Curvature of a Surface 164 4. Indicatrix of the Normal Curvature 165 5. Conjugate Coordinate Lines on a Surface 1t37 6. Lines of Curvature 168 7. Mean and Gaussian Curvature of a Surface 170 8. Example of a Surface of Constant Negative Gaussian Curvature 172 Exercises to Chapter X I 173 Chapter XII. Intrinsic Geometry of Surface 175 1. Gaussian Curvature as an Object of the Intrinsic Geo- metry of Surfaces 175 2. Geodesic Lines on a Surface 178 3. Extremal Property of Geodesics 179 4. Surfaces of Constant Gaussian Curvature 180 5. Gauss-Bonnet Theorem 181 6. Closed Surfaces 182 Exercises to Chapter XII 184 Part Three. Foundations of Geometry 186 Chapter XIII. Historical Survey 186 1. Euclid's Elements 186 2. Attempts to Prove the Fifth Postulate 1.88