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

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}
 * $\text {I}$


 * 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
 * $\text {II}$


 * 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
 * $\text {III}$


 * 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
 * $\text {IV}$


 * 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
 * $\text {V}$


 * 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
 * $\text {VI}$


 * APPENDIX
 * MISCELLANEOUS EXAMPLES $\text {I}$.
 * MISCELLANEOUS EXAMPLES $\text {II}$.
 * INDEX