Analytical Geometry

Front Cover
Title Page
Contents
Introduction
Chapter 1 Rectangular Cartesian Coordinates in a Plane
Sec. 1-1. Introducing Coordinates in a Plane
Sec. 1-2. The Distance Between Points
Sec, 1-3. Dividing a Line Segment in a Given Ratio
Sec. 1-4. The Notion of the Equation of a Curve. The Equation of a Circle
Sec. 1-5. The Equation of a Curve Represented Parametrically
Sec. 1-6. The Points of Intersection of Curves
Chapter 2 The Straight Line
Sec. 2-1. The General Equation of a Straight Line
Sec. 2-2. Particular Cases of the Equation of a Straight Line
Sec. 2-3. The Equation of a Straight Line in the Form Solved with Respect to y . The Angle Between Two Straight Lines
Sec. 2-4. The Parallelism and Perpendicularity Conditions of Two Straight Lines
Sec. 2-5. The Mutual Positions of a Straight Line and a Point. The Equation of a Straight Line in the Normal Form
Sec. 2-6. Basic Problems on the Straight Line
Sec. 2-7. Transformation of Coordinates
Chapter 3 Conic Sections
Sec. 3-1. Polar Coordinates
Sec. 3-2. Conic Sections and Their Equations in Polar Coordinates
Sec. 3-3. The Equations of Conic Sections in Rectangular Cartesian Coordinates in Canonical Form
Sec. 3-4. Studying the Shape of Conic Sections
Sec. 3-5. A Tangent Line to a Conic Section
Sec. 3-6. The Focal Properties of Conic Sections
Sec. 3-7. The Diameters of a Conic Section
Sec. 3-8. Second-Order Curves (Quadric Curves)
Chapter 4 Vectors
Sec. 4-1. Addition and Subtraction of Vectors
Sec. 4-2. Multiplication of a Vector by a Number
Sec. 4-3. Scalar Product of Vectors
Sec. 4-4. The Vector Product of Vectors
Sec. 4-5. The Triple Product of Vectors
Sec. 4-6. The Coordinates of a Vector Relative to a Given Basis
Chapter 6 A Plane and a Straight Line
Sec. 6-1. The Equation of a Plane
Sec. 6-2. Special Cases of the Position of a Plane Relative to a Coordinate System
Sec. 6-3. The Normal Form of the Equation of a Plane
See. 6-4. Relative Position of Planes
Sec. 6-5* Equations of the Straight Line
Sec. 6-6. Relative Position of a Straight Line and a Plane, of Two Straight Lines
Sec. 6-7. Basic Problems on the Straight Line and the Plane
Chapter 7 Surfaces of the Second Order (Quadric Surfaces)
Sec. 7-1. A Special System of Coordinates
Sec. 7-2. Quadric Surfaces Classified
Sec. 7-3. The Ellipsoid
Sec. 7-4. Hyperboloids
Sec. 7-5. Paraboloids
Sec. 7-6. The Cone and Cylinders
Sec. 7-7. Rectilinear Generators on Quadric Surfaces
Sec, 7-8. Diameters and Diametral Planes of a Quadric Surface
Chapter 8 Investigating Quadric Curves and Surfaces Specified by Equations of the General Form
Sec. 8-1. Transformation of the Quadratic Form to New Variables
Sec. 8-2. Invariants of the Equations of Quadric Curves and Surfaces with Reference to Transformation of Coordinates
Sec, 8-3. Investigating a Quadric Curve by Its Equation in Arbitrary Coordinates
Sec. 8-4. Investigating a Quadric Surface Specified by an Equation in Arbitrary Coordinates
Sec. 8-5. Diameters of a Curve, Diametral Planes of a Surface. The Centre of a Curve and a Surface
Sec. 8-6. Axes of Symmetry of a Curve. Planes of Symmetry of a Surface
Sec. 8-7. The Asymptotes of a Hyperbola. The Asymptotic Cone of a Hyperboloid
Sec. 8-8. A Tangent Line to a Curve. A Tangent Plane to a Surface
Chapter 9 Linear Transformatio
Sec. 9-1. Orthogonal Transformations
Sec. 9-2. Affine Transformations
Sec. 9-3. The Affine Transformation of a Straight Line and a Plane
Sec. 9-4. The Principal Invariant of the Affine Transformation
Sec. 9-5. Affine Transformations of Quadric Curves and Surfaces
Sec. 9-6. Projective Transformations
Sec. 9-7. Homogeneous Coordinates. Supplementing a Plane and a Space with Elements at Infinity.
Sec. 9-8. The Projective Transformations of Quadric Curves and Surfaces
Sec, 9-9. The Pole and PoJar
Sec, 9-10. Tangential Coordinates
Answers to the Exercises,Hints and Solutions
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9