Analytical Quadrics - 1st Edition - ISBN: 9780080136264, 9781483138299

Analytical Quadrics, Volume 14

1st Edition

Editors: I. N. Sneddon S. Ulam M. Stark
Authors: Barry Spain
eBook ISBN: 9781483138299
Imprint: Pergamon
Published Date: 1st January 1960
Page Count: 146
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Table of Contents


I. 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


41.Centre of A Section

42.Axes of Central Plane Section

43.Axes of Non-Central Plane Section

VI. Paraboloid


45.Polar Properties


47.Generators of Hyperbolic Paraboloid


49.Centre of A Section

50.Axes of Plane Section

VII. Homogeneous Coordinates

51.Homogeneous Coordinates

52.Point At Infinity

53.Straight Line


55.Intersection of Three Planes

56.Circle At Infinity

57.Circular Sections of Central Quadric


VIII. General Quadric


60.Cofactor Notation

61.Conjugate Points

62.Pole and Polar

63.Tangent Plane

64.Enveloping Cone


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 Δ

IX. 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 Coordinates

X. 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


I. Matrices

II. Solutions



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.


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Ratings and Reviews

About the Editors

I. N. Sneddon Editor

S. Ulam Editor

M. Stark Editor

About the Authors

Barry Spain Author