Spain / Sneddon / Ulam Analytical Quadrics

1. Auflage 2014
ISBN: 978-1-4831-3829-9
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 14, 146 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

Analytical Quadrics focuses on the analytical geometry of three dimensions. The book first discusses the theory of the plane, sphere, cone, cylinder, straight line, and central quadrics in their standard forms. The idea of the plane at infinity is introduced through the homogenous Cartesian coordinates and applied to the nature of the intersection of three planes and to the circular sections of quadrics. The text also focuses on paraboloid, including polar properties, center of a section, axes of plane section, and generators of hyperbolic paraboloid. The book also touches on homogenous coordinates. Concerns include intersection of three planes; circular sections of central quadric; straight line; and circle at infinity. The book also discusses general quadric and classification and reduction of quadric. Discussions also focus on linear systems of quadrics and plane-coordinates. The text is a valuable reference for readers interested in the analytical geometry of three dimensions.

Autoren/Hrsg.

Spain, Barry

Sneddon, I. N.

Ulam, S.

Stark, M.

Inhaltsverzeichnis

1;Front Cover;1
2;Analytical Quadrics;4
3;Copyright Page;5
4;Table of Contents;8
5;PREFACE;6
6;CHAPTER I. INTRODUCTION;12
6.1;1. Point coordinates;12
6.2;2. Distance between two points;13
6.3;3. Joachimsthal's section formula;14
6.4;4. Direction-cosines;15
6.5;5. Direction-ratios;16
6.6;6. Angle between two straight lines;16
6.7;7. Projections;18
7;CHAPTER II. PLANE—STRAIGHT LINE;20
7.1;8. Equation of a plane;20
7.2;9. Equations of a straight line;21
7.3;10. Straight line and plane;23
7.4;11. Sign of the expression u = .x+µ.+vz+p;24
7.5;12. Distance from a point to a plane;25
7.6;13. Pair of planes;26
7.7;14. Bundle of planes;26
7.8;15. Shortest distance between two skew lines;27
7.9;16. Coplanar lines;28
7.10;17. Rectangular axes related to two skew lines;29
8;CHAPTER III. SPHERE;31
8.1;18. Equation of sphere;31
8.2;19. Conjugate points;32
8.3;20. Pole and polar;34
8.4;21. Tangent plane and normal line;34
8.5;22. Angle of intersection of two spheres;36
8.6;23. Radical plane;36
8.7;24. Coaxal spheres;37
9;CHAPTER IV. QUADRIC CONE—CYLINDER;40
9.1;25. Quadric cone;40
9.2;26. Cone with vertex at the origin;40
9.3;27. Circular cone;41
9.4;28. Conditions that .(x,y,z) be a circular cone;42
9.5;29. Angle between two generators;42
9.6;30. Reciprocal cone;44
9.7;31. Rectangular and orthogonal cones;45
9.8;32. Quadric cylinder;45
10;CHAPTER V. CENTRAL QUADRIC;47
10.1;33. Central quadric;47
10.2;34. Ellipsoid;48
10.3;35. Hyperboloid of one sheet;49
10.4;36. Hyperboloid of two sheets;50
10.5;37. Polar properties;50
10.6;38. Generators;52
10.7;39. Generators of hyperboloid of one sheet;53
10.8;40. Normals;54
10.9;41. Centre of a section;55
10.10;42. Axes of central plane section;56
10.11;43. Axes of non-central plane section;57
11;CHAPTER VI. PARABOLOID;59
11.1;44. Paraboloid;59
11.2;45. Polar properties;61
11.3;46. Generators;61
11.4;47. Generators of hyperbolic paraboloid;62
11.5;48. Normals;63
11.6;49. Centre of a section;63
11.7;50. Axes of plane section;64
12;CHAPTER VII. HOMOGENEOUS COORDINATES;66
12.1;51. Homogeneous coordinates;66
12.2;52. Point at infinity;66
12.3;53. Straight line;66
12.4;54. Plane;68
12.5;55. Intersection of three planes;69
12.6;56. Circle at infinity;70
12.7;57. Circular sections of central quadric;72
12.8;58. Paraboloid;73
13;CHAPTER VIII. GENERAL QUADRIC;76
13.1;59. Quadric;76
13.2;60. Cofactor notation;77
13.3;61. Conjugate points;77
13.4;62. Pole and polar;78
13.5;63. Tangent plane;79
13.6;64. Enveloping cone;79
13.7;65. Generators;80
13.8;66. Polar of a line;81
13.9;67. Conjugate lines;82
13.10;68. Conjugate diameters;82
13.11;69. Principal diametral planes;84
13.12;70. Discriminating cubic;85
13.13;71. Circular sections;86
13.14;72. Rank of matrix .;87
14;CHAPTER IX. TRANSFORMATION OF AXES—INVARIANTS;89
14.1;73. Translation of axes;89
14.2;74. Invariants of a quadratic form under translation;90
14.3;75. Rotation of axes;91
14.4;76. Invariants of quadratic form under rotation;93
14.5;77. General transformation of coordinates;94
15;CHAPTER X. CLASSIFICATION AND REDUCTION OF QUADRIC;95
15.1;78. Equation of quadric ;95
15.2;79. Classification of quadrics ;96
15.3;80 Central quadric and cone;98
15.4;81. Paraboloid;99
15.5;82. Elliptic and hyperbolic cylinder;100
15.6;83. Parabolic cylinder and parallel plane-pair;101
16;CHAPTER XI. FOCI—CONFOCALS;104
16.1;84. Focus;104
16.2;85. Foci of central quadric;105
16.3;86. Foci of paraboloid;107
16.4;87. Confocal central quadrics;108
16.5;88. Confocal paraboloids;109
17;CHAPTER XII. LINEAR SYSTEMS OF QUADRICS;112
17.1;89. Pencil of quadrics;112
17.2;90. Self-polar tetrahedron of a pencil of quadrics;113
17.3;91. Contact of quadrics;114
17.4;92. Associated points;115
18;CHAPTER XIII. PLANE-COORDINATES;117
18.1;93. Plane-coordinates;117
18.2;94. Principle of duality;117
18.3;95. Quadric envelope;118
18.4;96. Conic;119
18.5;97. Cone;121
18.6;98. Joachimsthal's quadratic equation;121
18.7;99. Range of quadrics;123
18.8;100. Confocal quadrics;123
19;APPENDICES;126
19.1;I. Matrices;126
19.2;II. Solutions;128
20;INDEX;146