We now come to Analytical Geometry by* A. V. Pogorelov.*

Analytical geometry has no strictly defined contents. It is the method but not the subject under investigation, that constitutes the leading feature of this branch of geometry. The essence of this method consists in that geometric objects are associated in some standard way with equations (or systems of equations) so that geometric relations of figures are expressed through properties of their equations. For instance, in case of Cartesian coordinates any straight line in the plane is uniquely associated with a linear equation ax+by+ c = 0. The intersection of three straight, lines at one point is expressed by the condition of compatibility of a system of three equations which specify these lines.

Due to a multi purpose approach to solving various problems, the method of analytic geometry has become the leading method in geometric investigations and is widely applied in other fields of exact natural sciences, such as mechanics and physics. Analytical geometry joined geometry with algebra and analysis – the fact which has told fruitfully on further development of these three subject of mathematics. The principal ideas of analytical geometry are traced back to the French mathematician, Rene Descartes (1595-1650), who in 1637 described the fundamentals of its method in his famous work “Geometric”.

The present book, which is a course of lectures, treats the

fundamentals of the method of analytic geometry as applied to the

simplest geometric objects. It is designed for the university

students majoring in physics and mathematics,

This book was translated from the Russian by *Leonid Levant* and

was first published by *Mir Publishers* in 1980.

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** Table of Contents**

**Chapter 1 **

** Rectangular Cartesian Coordinates In a Plane 11**

Sec. 1-1. Introducing Coordinate in a Plane 11

Sec. 1-2. The Distance Between Points 15

Sec. 1-3. Dividing n Line Segment in a Given Ratio 17

Sec. 1-4. The Notion of the Equation of a Curve The Equation of a

Circle 21

Sec. 1-5. The Equation of a Curve Represented Parametrically 25

Sec. 1-6. The Points of Intersection of Curves 28

**Chapter 2 **

** The Straight Line 32**

Sec. 2-1. The General Equation of the Straight Line 32

Sec. 2-2. Particular Cases of the Equation of a Straight Line 35

Sec. 2-3. The Equation of a Straight. Line in the Form Solved

with Respect to y. The Angle Between Two Straight Lines 38

Sec. 2-4. The Parallelism and Perpenrlicularitp Conditions of Two

Straight Lines 40

Sec. 2-5. The Mutual Positions of a Straight Line and a Point 43

Sec. 2-6. Basic Problems on the Straight Line 47

Sec. 2-7. Transformation of Coordinates 49

**Chapter 3**

** Conic Sections 55 **

Sec. 3-1. Polar Coordinates 55

Sec. 3-2. Conic Sections and Their Equations in Polar Coordinates 58

Sec. 3-3. The Equations of Conic Sections in Rectangular Cartesian

Coordinates in Canonical Form 62

Sec. 3-4. Studying the Shape of Conic Sections 64

Sec. 3-5. A Tangent Line to a Conic Section 70

Sec 3-6. The Focal Properties of Conic Sections 74

Sec. 3-7. The Diameters of a Conic Section 78

Sec. 3-8. Second-Order Curves (Quadric Curves) 82

**Chapter 4 **

** Vectors 87**

Sec. 4-1. Addition and Subtraction of Vector’s 87

Sec. 4-2. Multiplication of a Vector by a Number 90

Sec. 4-3. Scalar Product of Vectors 93

Sec. 4-4. The Vector Product of Vector 96

Sec. 4-5. The Triple Product of Vectors 98

Sec. 4-6. The Coordinates of a Vector Relative to a Given Basis 101

**Chapter 5**

** Rectangular Cartesian Coordinates in Space 106**

Sec. 5-1. Cartesian Coordinates 106

Sec. 5-2. Elementary Problems of Solid Analytic Geometry 108

Sec. 5-3. Equations of a Surf’ace aml a Curve in Space 111

Sec. 5-4. Transformation of Coordinates 115

Chapter 6

A Plane and a Straight Line 119

Sec. 6-1. The Equation of a Plane 119

Sec. 6-2. Special Cases of the Position of a Plane Relative to a Coordinate System 121

Sec. 6-3. The Normal Form of the Equation of a Plane 123

Sec. 6-4. Relative Position of Planes 125

Sec. 6-5. Equations of the Straight Line 129

Sec. 6-6. Relative Position of a Straight Line and a Plane, of Two

Straight Lines 131

Sec. 6.7 Basic Problems on the Straight Line and the Plane 134

**Chapter 7**

** Surfaces of the Second Order (Quadric Surfaces)**

Sec. 7-1. A Special System of Coordinates 139

Sec. 7-2. Quadric Surfaces Classified 142

Sec. 7-3. The Ellipsoid 145

Sec. 7-4. Hyperboloids 148

Sec. 7-5. Paraboloids 150

Sec. 7-6. The Cone and Cylinders 152

Sec. 7-7. Rectilinear Generators on Quadric Surfaces 155

Sec. 7-8. Diameters and Diametral Planes of a Quadric Surface 157

**Chapter 8**

** Investigating Quadric Curves and Surfaces Specified by Equations of**

** the General Form 160**

Sec. 8-1. Transformation of the Quadratic Form to New Variables 160

Sec. 8-2. lnvariants of the Equations of Quadric Curves and Surfaces

with Reference to Transformation of Coordinates 162

Sec. 8-3. Investigating a Quadric Curve by Its Equation in Arbitrary

Coordinates 165

Sec. 8-4. Investigating s Quadric Surface Specified by an Equation in Arbitrary Coordinates 168

Sec. 8-5. Diameters of a Curve, Diametral Planes of a Surface. The Centre of a Curve and a Surface 171

Sec. 8-6. Axes of Symmetry of a Curve. Planes of Symmetry of a Surface 173

Sec. 8-7. The Asymptotes of a Hyperbola. The Asymptotic Cone of a Hyperboloid 175

Sec. 8-8. Tangent Line to a Curve. A Tangent Plane to a Surface 176

**Chapter 9**

** Linear Transformations 180**

Sec. 9-1. Orthogonal Transformation 180

Sec. 9-2. Affine Transformations 183

Sec. 9-3. The Affine Transformation of a Straight Line and a Plane 185

Sec. 9-4. The Principal Invariant of the Affine Transformation 187

Sec. 9-5. Affine Transformations of Quadric Curves and Surfaces 188

Sec. 9-6. Projective Transformations 192

Sec. 9-7. Homogeneous Coordinates. Supplementing a Plane and a Space with Elements at

Infinity 195

Sec. 9-8. The Projective Transformations of Quadric, Curves and Surfaces 198

Sec. 9-9. The Pole and Polar 201

Sec. 9-10. Tangential Coordinates 206

**Answers to the Exercises,Hints and Solutions 211**

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