ABOUT THE BOOKThis is a manual for the students of universities and teachers' training colleges. Containing the compulsory course of geometry, its particular impact is on elementary topics. The book is, therefore, aimed at professional 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 applying algebraic methods to elementary geometry.The second part, differential geometry, contains the basics of the theory of curves and surfaces. The third part, foundations of geometry, is original. The fourth part is devoted to certain topics of elementarygeometry. The book as a whole must interest the reader in school or university teacher's profession.The book was translated from the Russian by Leonid Levant, Aleksandr Repyev and Oleg Efimov and published by Mir in 1987.All credits to the original uploader.ContentsPreface 10Part One. Analytic Geometry 11Chapter I. Rectangular Cartesian Coordinates in the Plane 111. Introducing Coordinates in the Plane 112. 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 21Chapter II. Vectors in the Plane 261. Translation 262. Modulus and the Direction of a Vector 283. Components of a Vector 304. Addition of Vectors 305. Multiplication of a Vector by a Number 316. Collinear Vectors 327. Resolution of a Vector into Two Non-Collinear Vectors 33 8. Scalar Product 34 Exercises to Chapter II 36Chapter III. Straight Line in the Plane 381. 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 425. Normal Form of the Equation of a Straight Line 436. Transformation of Coordinates 447. Motions in the Plane 47 8. Inversion 47Exercises to Chapter IIIChapter IV. Conic Sections 531. Polar Coordinates 53 2. Conic Sections 54 3. Equations of Conic Sections in Polar Coordinates 564. Canonical Equations of Conic Sections in Rectangular 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 71Chapter V. Rectangular Cartesian Coordinates and Vectors in Space 761. 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. Transformation of Coordinates 85 9. Equations of a Surface and a Curve in Space 87Exercises to Chapter V 89Chapter VI.Plane and a Straight Line in Space 951. 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 103Chapter VII. Quadric Surfaces 1091. 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 123Part Two.Differential Geometry 126Chapter VIII. Tangent and Osculating Planes of Curve 1261. Concept of Curve 126 2. Regular Curve 127 3. Singular Points of a Curve 128 4. Vector Function of Scalar Argument 1295. 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 136Exercises to Chapter VIII 137Chapter IX. Curvature and Torsion of Curve 1401. 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 148Exercises to Chapter IX 149Chapter X. Tangent Plane and Osculating Paraboloid of Surface 1511. 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 158Exercises to Chapter X 159Chapter XI. Surface Curvature 1611. 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 167 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 XI 173Chapter XII. Intrinsic Geometry of Surface 1751. Gaussian Curvature as an Object of the Intrinsic Geometry of Surfaces 1752. Geodesic Lines on a Surface 1783. Extremal Property of Geodesics 179 4. Surfaces of Constant Gaussian Curvature 180 5. Gauss-Bonnet Theorem 1816. Closed Surfaces 182Exercises to Chapter XII 184Part Three. Foundations of Geometry 186Chapter XIII. Historical Survey 1861. Euclid's Elements 186 2. Attempts to Prove the Fifth Postulate 1883. Discovery of Non-Euclidean Geometry 189 4. Works on the Foundations of Geometry in the Second Half of the 19th century 191 5. System of Axioms for Euclidean Geometry according to D. Hilbert 192Chapter XIV. System of Axioms for Euclidean Geometry and Their Immediate Corollaries 1941. Basic Concepts 194 2. Axioms of Incidence 195 3. Axioms of Order 196 4. Axioms of Measure for Line Segments and Angles 197 5. Axiom of Existence of a Triangle Congruent to a Given One 199 6. Axiom of Existence of a Line Segment of Given Length 200 7. Parallel Axiom 202 8. Axioms for Space 202Chapter XV. Investigation of Euclidean Geometry Axioms 2031. Preliminaries 203 2. Cartesian Model of Euclidean Geometry 204 3. "Betweenness" Relation for Points in a Straight Line. Verification of the Axioms of Order 205 4. Length of a Segment. Verification of the Axiom of Measure for Line Segments 207 5. Measure of Angles in Degrees. Verification of Axiom III* 208 6. Validity of the Other Axioms in the Cartesian Model 210 7. Consistency and Completeness of the Euclidean Geometry Axiom System 212 8. Independence of the Axiom of Existence of a Line Segment of Given Length 214 9. Independence of the Parallel Axiom 216 10. Lobachevskian Geometry 218Chapter XVI. Projective Geometry 2221. Axioms of Incidence for Projective Geometry 222 2. Desargues Theorem 223 3. Completion of Euclidean Space with the Elements at Infinity 225 4. Topological Structure of a Projective Straight Line and Plane 226 5. Projective Coordinates and Projective Transformations 228 6. Cross Ratio 230 7. Harmonic Separation of Pairs of Points 232 8. Curves of the Second Degree and Quadric Surfaces 233 9. Steiner Theorem 23510. Pascal Theorem 236 11. Pole and Polar 238 12. Polar Reciprocation. Brianchon Theorem 240 13. Duality Principle 241 14. Various Geometries in Projective Outlook 243Exercises to Chapter XVI 245Part Four. Certain Problems of Elementary Geometry 247Chapter XVII. Methods for Solution of Construction Problems 2471. Preliminaries 247 2. Locus Method 248 3. Similarity Method 250 4. Reflection Method 251 5. Translation Method 251 6. Rotation Method 252 7. Inversion Method 253 8. On Solvability of Construction Problems 255Exercises to Chapter XVII 256Chapter XVIII. Measuring Lengths, Areas and Volumes 2581. Measuring Line Segments 258 2. Length of a Circumference 260 3. Areas of Figures 261 4. Volumes of Solids 265 5. Area of a Surface 267Chapter XIX. Elements of Projection Drawing 2681. Representation of a Point on an Epure 268 2. Problems Leading to a Straight Line 269 3. Determination of the Length of a Line Segment 270 4. Problems Leading to a Straight Line and a Plane 271 5. Representation of a Prism and a Pyramid 273 6. Representation of a Cylinder, a Cone and a Sphere 274 7. Construction of Sections 275Exercises to Chapter XIX 277Chapter XX. Polyhedral Angles and Polyhedra 2781. Cosine Law for a Trihedral Angle 2782. Trihedral Angle Conjugate to a Given One 2793. Sine Law for a Trihedral Angle 2804. Relation Between the Face Angles of a Polyhedra Angles 2815. Area of a Spherical Polygon 2826. Convex Polyhedra. Concept of Convex Body 2837. Euler Theorem for Convex Polyhedra 284 8. Cauchy Theorem 285 9. Regular Polyhedra 288Exercises to Chapter XX 289 Answers to Exercises, Hints and Solutions 291