Book:Robert J.T. Bell/Coordinate Solid Geometry

Robert J.T. Bell: Coordinate Solid Geometry

Published $\text {1938}$, Macmillan and Co., Limited

Reprint of chapters I to IX of An Elementary Treatise on Coordinate Geometry of Three Dimensions from 1911.

Subject Matter

• Solid Geometry

Contents

Publisher's Note (May, 1938)

Chapter $\text {I}$: Systems of Coordinates. The Equation to a Surface

1. Segments

2. Relations between collinear segments

3. Cartesian coordinates

4. Sign of direction of rotation

5. Cylindrical coordinates

6. Polar coordinates

7. Change of origin

8. Point dividing line in given ratio

9. The equation to a surface

10. The equation to a curve

11. Surfaces of revolution

Chapter $\text {II}$: Projections. Direction-Cosines. Direction-Ratios

12. The angles between two directed lines

13. The projection of a segment

14. Relation between a segment and a projection

15. The projection of a broken line

16. The angle between two planes

17. Relation between areas of a triangle and its projection

18. Relation between areas of a polygon and its projection

19. Realtion between areas of a closed curve and its projection

20. Direction-cosines -- definition

21, 22. Direction-cosines (axes rectangular)

23. The angle between two lines with given direction-cosines

24. Distance of a point from a line

25, 26. Direction-cosines (axes oblique)

27. The angle between two lines with given direction-cosines

28, 29, 30. Direction-ratios

31. Relation between direction-cosines and direction-ratios

32. The angle between two lines with given direction-ratios

Chapter $\text {III}$: The Plane. The Straight Line. The Volume of a Tetrahedron.

33. Forms of the equation to a plane

34, 35. The general equation to a plane

36. The plane through three points

37. The distance of a point from a plane

38. The planes bisecting the angles between two given planes

39. The equations to a straight line

40. Symmetrical form of equations

41. The line through two given points

42. The direction-ratios found from the equations

43. Constants in the equations to a line

44. The plane and the straight line

45. The intersection of three planes

46. Lines intersecting two given lines

47. Lines intersecting three given lines

48. The condition that two given lines be coplanar

49. The shortest distance between two given lines

50. Problems relating to two given non-intersecting lines

51. The volume of a tetrahedron

Chapter $\text {IV}$: Change of Axes

52. Formulae of transformation (rectangular axes)

53. Relations between the directin-cosines of three mutually perpendicular lines

54. Transformation to examine the section of a given surface by a given plane

55. Formulae of transformation (oblique axes}

Examples $\text {I}$

Chapter $\text {V}$: The Sphere

56. The equation to a sphere

57. Tangents and tangent plane to a sphere

58. The radical plane of two spheres

Examples $\text {II}$

Chapter $\text {VI}$: The Cone

59. The equation to a cone

60. The angle between the lines in which a plane cuts a cone

61. The condition of tangency of a plane and a cone

62. The condition that a cone has three mutually perpendicular generators

63. The equation to a cone with a given base

Examples $\text {III}$

Chapter $\text {VII}$: The Central Conicoids. The Cone. The Paraboloids

64. The equation to a central conicoid

65. Diametral planes and conjugate diameters

66. Points of intersection of a line and a conicoid

67. Tangents and tangent planes

68. Condition that a plane should touch a conicoid

69. The polar plane

70. Polar lines

71. Section with a given centre

72. Locus of the mid-points of a system of parallel chords

73. The enveloping cone

74. The enveloping cylinder

75. The normals

76. The normals from a given point

77. Conjugate diameters and diametral planes

78. Properties of the cone

79. The equation of a paraboloid

80. Conjugate diametral planes

81. Diameters

82. Tangent planes

83. Diametral planes

84. The normals

Examples $\text {IV}$

Chapter $\text {VIII}$: The Axes of Plane Sections. Circular Sections.

85. The determination of axes

86. Axes of a central section of a central conicoid

87. Axes of any section of a central conicoid

88. Axes of a section of a paraboloid

89. The determination of circular sections

90. Circular sections of the ellipsoid

91. Any two circular sections from opposite systems lie on a sphere

92. Circular sections of the hyprboloids

93. Circular sections of the general central conicoid

94. Circular sections of the paraboloids

95. Umbilics

Examples $\text {V}$

Chapter $\text {IX}$: Generating Lines

96. Ruled surfaces

97. The section of a surface by a tangent plane

98. Line meeting conicoid in three points of a generator

99. Conditions that a line should be a generator

100. System of generators of a hyperboloid

101. Generators of same system do not intersect

102. Generators of opposite systems intersect

103. Locus of points of intersection of perpendicular generators

104. The projections of generators

105. Along a generator $\theta \pm \phi$ is constant

106. The systems of generators of the hyperbolic paraboloid

107. Conicoids through three given lines

108. General equation to conicoid through two given lines

109. The equation to the conicoid through three given lines

110, 111. The straight lines which meet four given lines

112. The equation to a hyperboloid when generators are coordinate axes

113. Properties of a given generator

114. The central point and parameter of distribution

Examples $\text {VI}$

APPENDIX

MISCELLANEOUS EXAMPLES $\text {I}$.

MISCELLANEOUS EXAMPLES $\text {II}$.

INDEX