Pogorelov – Analytical Geometry

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

analytical geometry

  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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One Response to Pogorelov – Analytical Geometry

  1. m95 says:

    very nice post thanks a lot

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