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B]<OU 160228 >M z </)> ro^ 160228 >m >*B]<OU ^S z ^g OSMANIA UNIVERSITY LIBRARY >5T /6 Call No. ^ _ . Accession No. Author Title This book should be returnedion or before the^date last marked below. H. B. E*HMOB KPATKHfi KYPC AHAJIHTHMECKOfi FEOMETPHM FOCy/lAPCTBEHHOE 4>H3HKO-MATEMATH4ECKOft ^HTEPATVPbl M o c K B a N. YEFIMOV A Brief Course in Analytic Geometry TRANSLATED FROM THE RUSSIAN by O. S R O K A PEACE PUBLISHERS MOSCOW Written by Professor N. Yefimov, Dr. Phys. Math. Sc., this book presents, in concise form, the theoretical foundations of plane and solid analytic geometry. Also, an elementary outline of the theory of determinants is given in the Appendix. The textbook is intended for students of higher educational institutions and for engineers engaged in the field of quadric surface design. The book is illustrated with 122 drawings. CONTENTS Part One PLANE ANALYTIC GEOMETRY Chapter 1. Coordinates on a Straight Line and in a Plane 11 1. An Axis and Segments of an Axis 11 2. Coordinates on a Line. The Number Axis 14 3. Rectangular Cartesian Coordinates in a Plane. A Note on Ob- lique Cartesian Coordinates 16 4. Polar Coordinates 20 Chapter 2. Elementary Problems of Plane Analytic Geometry 23 5. Projection of a Line Segment. Distance Between Two Points . 23 6. Calculation of the Area of a Triangle 28 7. Division of a Line Segment in a Given Ratio 30 8. Transformation of Cartesian Coordinates by Translation of Axes 35 9. Transformation of Rectangular Cartesian Coordinates by Rotation of Axes 36 10. Transformation of Rectangular Cartesian Coordinates by Change of Origin and Rotation of Axes 38 Chapter 3. The Equation of a Curve . 41 11. The Concept of the Equation of a Curve. Examples of Curves Represented by Equations 41 12. Examples of Deriving the Equation of a Given Curve 48 13. The Problem of the Intersection of Two Curves 50 14. Parametric Equations of a Curve 51 15. Algebraic Curves - . 52 Chapter 4. Curves of the First Order 55 16. The Slope of a Straight Line 55 17. The Slope-intercept Equation of a Straight Line 56 18. Calculation of the Angle Between Two Straight Lines. Conditions for the Parallelism and Perpendicularity of Two Straight Lines . 59 19. The Straight Line As the Curve of the First Order. The General Equation of the Straight Line 61 20. Incomplete Equations of the First Degree. The Intercept Equation of a Straight Line 63 21. Discussion of a System of Equations Representing Two Straight Lines .65 22. The Normal Equation of a Straight Line. The Problem of Cal- culating the Distance of a Point from a Straight Line 68 23. The Equation of a Pencil of Lines 71 Chapter 5. Geometric Properties of Curves of the Second Order 75 24. The Ellipse. Definition of the Ellipse and Derivation of Its Canon- ical Equation 75 6 Contents 25. Discussion of the Shape of the Ellipse 79 26. The Eccentricity of the Ellipse 81 27. Rational Expressions for Focal Radii of the Ellipse 82 28. Point-by-point Construction of the Ellipse. The Parametric Equations of the Ellipse 83 29. The Ellipse as the Projection of a Circle on a Plane. The Ellipse as the Section of a Circular Cylinder by a Plane 84 30. The Hyperbola. Definition of the Hyperbola and Derivation of Its Canonical Equation 87 31. Discussion of the Shape of the Hyperbola .91 32. The Eccentricity of the Hyperbola 97 33. Rational Expressions for Focal Radii of the Hyperbola 98 34. The Directrices of the Ellipse and Hyperbola 99 35. The Parabola. Derivation of the Canonical Equation of the Pa- rabola .103 36. Discussion of the Shape of the Parabola 105 37. The Polar Equation of the Ellipse, Hyperbola and Parabola . 107 38. Diameters of Curves of the Second Order 109 39. The Optical Properties of the Ellipse, Hyperbola and Parabola 114 40. The Ellipse, Hyperbola and Parabola as Conic Sections .... 115 Chapter 6. Transformation of Equations by Change of Coordinates .... 116 41. Examples of Reducing the General Equation of a Second-Order Curve to Canonical Form 116 42. The Hyperbola as the Inverse Proportionality Graph. The Pa- rabola as the Graph of a Quadratic Function 125 Part Two SOLID ANALYTIC GEOMETRY 7. Chapter Some Elementary Problems of Solid Analytic Geometry . 131 43. Rectangular Cartesian Coordinates in Space 131 44. The Concept of a Free Vector. The Projection of a Vector on an Axis 134 45. The Projections of a Vector on the Coordinate Axes 138 46. Direction Cosines .141 47. Distance between Two Points. Division of a Line Segment in a Given Ratio 142 Chapter 8. Linear Operations on Vectors 143 48. Definitions of Linear Operations 143 49. Basic Properties of Linear Operations 144 50. The Vector Difference 147 51. Fundamental Theorems on Projections . 149 52. Resolution of Vectors into Components 152 Chapter 9. The Scalar Product of Vectors 157 53. The Scalar Product and Its Basic Properties 157 54. Representation of the Scalar Product in Terms of the Coordi- nates of the Vector Factors 160 Chapter 10. The Vector and Triple Scalar Products of Vectors 163 55. The Vector Product and Its Basic Properties 163 56. Representation of the Vector Product in Terms of the Coordinates of the Vector Factors 170 Contents 57. The Triple Scalar Product .172 58. Representation of the Triple Scalar Product in Terms of the Coordinates of the Vector Factors 176 Chapter 1L The Equation of a Surface and the Equations of a Curve . 178 59. The Equation of a Surface 178 60. The Equations of a Curve. The Problem of the Intersection of Three Surfaces 179 61. The Equation of a Cylindrical Surface with Elements Parallel to a Coordinate Axis 181 62. Algebraic Surfaces 183 Chapter 12. The Plane as the Surface of the First Order. The Equations of a Straight Line 185 63. The Plane as the Surface of the First Order 185 64. Incomplete Equations of Planes. The Intercept Form of the Equa- tion of a Plane 188 65. The Normal Equation of a Plane. The Distance of a Point from a Plane 190 66, The Equations of a Straight Line 194 67. The Direction Vector of a Straight Line. The Canonical Equations of a Straight Line. The Parametric Equations of a Straight Line . 198 68. Some Additional Propositions and Examples 201 Chapter 13 Quadric Surfaces 207 69 The Ellipsoid and the Hyperboloids 207 70. The Quadric Cone 213 71. The Paraboloids 215 72. The Quadric Cylinders 219 73. The Rectilinear Generators of the Hyperboloid of One Sheet. The Shukhov Towers 220 Appendix. The Elements of the Theory of Determinants 225 1. Determinants of the Second Order and Systems of Two Equations of the First Degree in Two Unknowns 225 2. A Homogeneous System of Two Equations of the First Degree in Three Unknowns 229 3. Determinants of the Third Order 232 4. Cofactors and Minors 236 5. Solution and Analysis of a System of Three First-degree Equa- tions in Three Unknowns 240 6. The Concept of a Determinant of Any Order 217 PART ONE Plane Analytic Geometry Chapter 1 COORDINATES ON A STRAIGHT LINE AND IN A PLANE 1. An Axis and Segments of an Axis 1. Consider an arbitrary straight line. It extends in two oppo- site directions. Let us choose at will one of these directions and refer to it as positive (and to the other direction as negative). A straight line to which a positive direction has been assigned is called an axis. In diagrams, the positive direction of an axis is indicated by an arrowhead (see, for example, Fig. 1, where an axis a is shown). B Fig. 1. * 2. Let there be given an axis, and let there also be specified a unit segment, that is, a unit of length by means of which any segment can be measured, so that the length of any segment may be determined. Take two arbitrary points on a given axis and denote them by the letters A and B. The segment bounded by the points A, B is called a directed segment if one of these points has been designated as the initial point, and the other as the terminal point of the segment. A segment is considered to be directed from its initial to its terminal point. In the succeeding pages, a directed segment is denoted by the two letters that mark the points bounding the segment; the letter marking the initial point is always written first, and a horizontal bar is placed over the letters. Thus, AB denotes the directed seg- ment bounded by the points A, B and having A as its initial point; BA denotes the directed segment bounded by the points A, B and having B as its initial point. When dealing with directed segments of an axis in our future work, we shall often refer to them simply as segments, omitting the word "directed". Coordinates on a Straight Line and in a Plane Let us now agree to define the value of a segment AB of an axis as the number equal to the length of the segment and taken with a plus or minus sign according as the direction of the seg- ment agrees with the positive or the negative_direction of the axis. We shall denote the value of a segment AB by the symbol AB (without the horizontal bar). We do not exclude the case when the points A and B coincide; in this case, the segment AB is called a zero segment, since its value is equal to zero. A zero segment has no definite direction, and hence the term "directed" may be applied to such a segment only conventionally.
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