Analytical Quadrics
- 1st Edition, Volume 14 - January 1, 1960
- Latest edition
- Author: Barry Spain
- Editors: I. N. Sneddon, S. Ulam, M. Stark
- Language: English
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… Read more
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.
ContentsI. Introduction 1. Point Coordinates 2. Distance Between Two Points 3. Joachimsthal's Section Formula 4. Direction-Cosines 5. Direction-Ratios 6. Angle Between Two Straight Lines 7. Projections II. Plane—Straight Line 8. Equation of A Plane 9. Equations of A Straight Line 10. Straight Line and Plane 11. Sign of the Expression u = λx + μy + vz + ρ 12. Distance From A Point to A Plane 13. Pair of Planes 14. Bundle of Planes 15. Shortest Distance Between Two Skew Lines 16. Coplanar Lines 17. Rectangular Axes Related to Two Skew Lines III. Sphere 18. Equation of Sphere 19. Conjugate Points 20. Pole and Polar 21. Tangent Plane and Normal Line 22. Angle of Intersection of Two Spheres 23. Radical Plane 24. Coaxal Spheres IV. Quadric Cone—Cylinder 25. Quadric Cone 26. Cone With Vertex At The Origin 27. Circular Cone 28. Conditions That ψ(x,y,z) Be A Circular Cone 29. Angle Between Two Generators 30. Reciprocal Cone 31. Rectangular and Orthogonal Cones 32. Quadric Cylinder V. Central Quadric 33. Central Quadric 34. Ellipsoid 35. Hyperboloid of One Sheet 36. Hyperboloid of Two Sheets 37. Polar Properties 38. Generators 39.Generators of Hyperboloid of One Sheet 40.Normals 41.Centre of A Section 42.Axes of Central Plane Section 43.Axes of Non-Central Plane SectionVI. Paraboloid 44.Paraboloid 45.Polar Properties 46.Generators 47.Generators of Hyperbolic Paraboloid 48.Normals 49.Centre of A Section 50.Axes of Plane SectionVII. Homogeneous Coordinates 51.Homogeneous Coordinates 52.Point At Infinity 53.Straight Line 54.Plane 55.Intersection of Three Planes 56.Circle At Infinity 57.Circular Sections of Central Quadric 58.ParaboloidVIII. General Quadric 59.Quadric 60.Cofactor Notation 61.Conjugate Points 62.Pole and Polar 63.Tangent Plane 64.Enveloping Cone 65.Generators 66.Polar of A Line 67.Conjugate Lines 68.Conjugate Diameters 69.Principal Diametral Planes 70.Discriminating Cubic 71.Circular Sections 72.Rank of Matrix ΔIX. Transformation of Axes—Invariants 73.Translation of Axes 74.Invariants of A Quadratic Form Under Translation 75.Rotation of Axes 76.Invariants of Quadratic Form Under Rotation 77.General Transformation of CoordinatesX. Classification and Reduction of Quadric 78.Equation of Quadric 79.Classification of Quadrics 80.Central Quadric and Cone 81. Paraboloid 82. Elliptic and Hyperbolic Cylinder 83. Parabolic Cylinder and Parallel Plane-Pair XI. Foci—Confocals 84. Focus 85. Foci of Central Quadric 86. Foci of Paraboloid 87. Confocal Central Quadrics 88. Confocal Paraboloids XII. Linear Systems of Quadrics 89. Pencil of Quadrics 90. Self-Polar Tetrahedron of A Pencil of Quadrics 91. Contact of Quadrics 92. Associated Points XIII. Plane-Coordinates 93. Plane-Coordinates 94. Principle of Duality 95. Quadric Envelope 96. Conic 97. Cone 98. Joachimsthal's Quadratic Equation 99. Range of Quadrics 100. Confocal Quadrics Appendices I. Matrices II. Solutions Index
- Edition: 1
- Latest edition
- Volume: 14
- Published: January 1, 1960
- Language: English
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