Geometry

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CONTENTS
PREFACE
Part One ANALYTIC GEOMETRY
Chapter I RECTANGULAR CARTESIAN COORDINATESIN THE PLANE
1. Introducing Coordinates in the Plane
2. Distance Between Two Points
3. Dividing a Line Segment in a Given Ratio
4. Equation of a Curve. Equation of a Circle
5. Parametric Equations of a Curve
6. Points of Intersection of Curves
7. Relative Position of Two Circles
EXERCISES TO CHAPTER I
Chapter II VECTORS IN THE PLANE
1. Translation
2. Modulus and the Direction of a Vector
3. Components of a Vector
4. Addition of Vectors
5. Multiplication of a Vector by a Number
6. Collinear Vectors
7. Resolution of a Vector into Two Non-Collinear Vectors
8. Scalar Product
EXERCISES TO CHAPTER II
Chapter III STRAIGHT LINE IN THE PLANE
1. Equation of a Straight Line.General Form
2. Position of a Straight Line Relative to a Coordinate System
3. Parallelism and Perpendicularity Condition for Straight Lines
4. Equation of a Pencil of Straight Lines
5. Normal Form of the Equation of a Straight Line
6. Transformation of Coordinates
7. Motions in the Plane
8. Inversion
EXERCISES TO CHAPTER III
Chapter IV CONIC SECTIONS
1. Polar Coordinates
2. Conic Sections
3. Equations of Conic Sections in Polar Coordinates
4. Canonical Equations of Conic Sections in Rectangular Cartesian Coordinates
5. Types of Conic Sections
6. Tangent Line to a Conic Section
7. Focal Properties of Conic Sections
8. Diameters of a Conic Section
9. Curves of the Second Degree
EXERCISES TO CHAPTER IV
Chapter V RECTANGULAR CARTESIAN COORDINATES AND VECTORS IN SPACE
1. Cartesian Coordinates in Space. Introduction
2. Translation in Space
3. Vectors in Space
4. Decomposition of a Vector into Three Non-coplanar Vectors
5. Vector Product of Vectors
6. Scalar Triple Product of Vectors
7. Affine Cartesian Coordinates
8. Transformation of Coordinates
9. Equations of a Surface and a Curve in Space
EXERCISES TO CHAPTER V
Chapter VI PLANE AND A STRAIGHT LINE IN SPACE
1. Equation of a Plane
2. Position of a Plane Relative to a Coordinate System
3. Normal Form of Equations of the Plane
4. Parallelism and Perpendicularity of Planes
5. Equations of a Straight Line
6. Relative Position of a Straight Line and a Plane, of Two Straight Lines
7. Basic Problems on Straight Lines and Planes
EXERCISES TO CHAPTER VI
Chapter VII QUADRIC SURFACES
1. Special System of Coordinates
2. Classification of Quadric Surfaces
3. Ellipsoid
4. Hyperboloids
5. Paraboloids
6. Cone and Cylinders
7. Rectilinear Generators on Quadric Surfaces
8. Diameters and Diametral Planes of a Quadric Surface
9. Axes of Symmetry for a Curve. Planes of Symmetry for a Surface
EXERCISES TO CHAPTER VII
Part Two DIFFERENTIAL GEOMETRY
Chapter VIII TANGENT AND OSCULATING PLANES OF CURVE
1. Concept of Curve
2. Regular Curve
3. Singular Polnts Of a Curve
4. Vector Function of Scalar Argument
5. Tangent to a Curve
6. Equations of Tangents for Various Methods of Specifying a Curve
7. Osculating Plane of a Curve
8. Envelope of a Family of Plane Curves
EXERCISES TO CHAPTER VIII
Chapter IX CURVATURE AND TORSION OF CURVE
1. Length of a Curve
2. Natural Parametrization of a Curve
3. Curvature
4. Torsion of a Curve
5. Frenet Formulas
6. Evolute and Evolvent of a Plane Curve
EXERCISES TO CHAPTER IX
Chapter X TANGENT PLANE AND OSCULATING, PARABOLOID OF SURFACE
1. Concept of Surface
2. Regular Surfaces
3. Tangent Plane to a Surface
4. Equation of a Tangent Plane
5. Osculating Paraboloid of a Surface
6. Classification of Surface Points
EXERCISES TO CHAPTER X
Chapter XI SURFACE CURVATURE
1. Surface Linear Element
2. Area of a Surface
3. Normal Curvature of a Surface
4. Indicatrix of the Normal Curvature
5. Conjugate Coordinate Lines on a Surface
6. Lines of Curvature
7. Mean and Gaussian Curvatul'e of a Surface
8. Example of a Surface of Constant Negative Gaussian Curvature
EXERCISES TO CHAPTER XI
Chapter XII INTRINSIC GEOMETRY OF SURFACE
1. Gaussian Curvature as an Object of the Intrinsic Geometry of Surfaces
2. Geodesic Lines on a Surface
3. Extremal Property of Geodesics
4. Surfaces of Constant Gaussian Curvature
5. Gauss-Bonnet Theorem
6. Closed Surfaces
EXERCISES TO CHAPTER XII
Part Three FOUNDATIONS OF GEOMETRY
Chapter XIII HISTORICAL SURVEY
1. Euclid's Elements
2. Attempts to Prove the Fifth Postulate
3. Discovery of Non-Euclidean Geometry
4. Works on the Foundations of Geometry in the Second Half of the 19th Century
5. System of Axioms for Euclidean Geometry according to D. Hilbert
Chapter XIV SYSTEM OF AXIOMS FOR EUCLIDEAN GEOMETRY AND THEIR IMMEDIATE COROLLARIES
1. Basic Concepts
2. Axioms of Incidence
3. Axioms of Order
4. Axioms of Measure for Line Segmentsand Angles
5. Axiom of Existence of a Triangle Congruent to ~ Given One
6. Axiom of Existence of a Line Segment of Given Length
7. Parallel Axiom
8. Axioms for Space :
Chapter XV INVESTIGATION OF EUCLIDEAN GEOMETRY AXIOMS
1. Preliminaries
2. Cartesian Model of Euclidean Geometry
3. "Betweenness" Relation for Points in a Straight Line. Verification of the Axioms of Order
4. Length of a Segment. V erifieation of the Axiom of Measure for Line Segments
5. Measure of Angles in Degrees. Verification of Axiom Ill2
6. Validity of the Other Axioms in the Cartesian Model
7. Consistency and Completeness of the Euclidean Geometry Axiom System
8. Independence of the Axiom of Existence of a Line Segment of Given Length
9. Independence of the Parallel Axiom
10. Lobachevskian Geometry
Chapter XVI PROJECTIVE GEOMETRY
1. Axioms of Incidence for Projective Geometry
2. Desargues Theorem
3. Completion of Euclidean Space with the Elements at Infinity
4. Topological Structure of a Projective Straight Line and Plane
5. Projective Coordinates and Projective Transformations
6. Cross Ratio
7. Harmonic Separation of Pairs of Points
8. Curves of the Second Degree and Quadric Surfaces
9. Steiner Theorem
10. Pascal Theorem
11. Pole and Polar
12. Polar Reciprocation. Brianchon Theorem
13. Duality Principle
14. Various Geometries in Projective Outlook
EXERCISES TO CHAPTER XVI
Part Four CERTAIN PROBLEMS OF ELEMENTARY GEOMETRY
Chapter XVII METHODS FOR SOLUTION OF CONSTRUCTION PROBLEMS
1. Preliminaries
2. Locus Method
3. Similarity Method
4. Reflection Method
5. Translation Method
6. Rotation Method
7. Inversion Method
8. On Solvability of Construction Problems
EXERCISES TO CHAPTER XVII *
Chapter XVIII MEASURING LENGTHS, AREAS AND VOLUMES
1. Measuring Line Segments
2. Length of a Circumference
3. Areas of Figures
4. Volumes of Solids
5. Area of a Surface
Chapter XIX ELEMENTS OF PROJECTION DRAWING
1. Representation of a Point on an Epure
2. Problems Leading to a Straight Line
3. Determination of the Length of a Line Segment
4. Problems Leading to a Straight Line and a Plane
5. Representation of a Prism and a Pyramid
6. Representation of a Cylinder, a Cone and a Sphere
7. Construction of Sections
EXERCISES TO CHAPTER XIX
Chapter XX POLYHEDRAL ANGLES AND POLYHEDRA
1. Cosine Law for a Trihedral Angle
2. Trihedral Angle Conjugate to a Given One
3. Sine Law for a Trihedral Angle
4. Relation Between the Face Angles of a Polyhedral Angle
5. Area of a Spherical Polygon
6. Convex Polyhedra. Concept of Convex Body
7. Euler Theorem for Convex Polyhedra
8. Cauchy Theorem
9. Regular Polyhedra
EXERCISES TO CHAPTER XX
ANSWERS TO EXERCISES, HINTS AND SOLUTIONS
Chapter I
Chapter II
Chapter III
Chapter IV
Chapter V
Chapter VI
Chapter VII
Chapter VIII
Chapter IX
Chapter X
Chapter XI
Chapter XII
Chapter XII
Chapter XVI
Chapter XVII
Chapter XIX
Chapter XX
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