Book:D.M.Y. Sommerville/Analytical Geometry of Three Dimensions

Subject Matter

 * Solid Geometry

Contents

 * Preface


 * Chap. $\text {I}$: Cartesian coordinate-system
 * 1.1. Cartesian coordinates
 * 1.2. Radius-vector, direction-angles
 * 1.3. Change of origin
 * 1.4. Distance between two points
 * 1.5. Angle between two lines, actual direction-cosines
 * 1.6. Perpendicularity and parallelism
 * 1.7. Position-ratio of a point w.r.t. two base-points
 * 1.8. General cartesian coordinates
 * 1.9. Examples


 * Chap. $\text {II}$: The straight line and plane
 * 2.1. Degrees of freedom
 * 2.2. Angles
 * 2.31. Intersection of a straight line and a plane
 * 2.41. Intersection of three planes
 * 2.5. Number of data which determine a point, a plane and a straight line
 * 2.6. Imaginary elements
 * 2.71. Distance from a point to a plane
 * 2.8. Volume of a tetrahedron
 * 2.9. Transformation of coordinates


 * Chap. $\text {III}$: General homogeneous or projective coordinates
 * 3.1. Projective geometry
 * 3.2. One-to-one correspondence
 * 3.3. Cross-ratio of four parameters
 * 3.41. Geometrical cross-ratio
 * 3.5. Homography, double-points
 * 3.6. Geometrical cross-ratio as a number
 * 3.71. Cross-ratios of different geometrical forms
 * 3.81. Transition from projective to metrical geometry
 * 3.91. Analytical representation of a homography
 * 3.95. Examples


 * Chap. $\text {IV}$: The sphere
 * 4.1. Equation in terms of centre and radius
 * 4.2. Power of a point w.r.t. a sphere
 * 4.3. Sphere through four given points
 * 4.41. Intersection of sphere and plane
 * 4.5. Pole and polar w.r.t. a sphere
 * 4.6. Linear systems of spheres
 * 4.7. Inversion in a sphere
 * 4.8. The circle at infinity
 * 4.9. Examples


 * Chap. $\text {V}$: The cone and cylinder
 * 5.1 Equation of a cone
 * 5.2 Intersection of a cone and a plane through the vertex
 * 5.3. Polar of a point w.r.t. a cone
 * 5.4. Reciprocal cones
 * 5.5. Rectangular generators
 * 5.6. Relation between geometry of cones and geometry of conics
 * 5.7. Cylinders
 * 5.9.Examples


 * Chap. $\text {VI}$: Types of surfaces of the second order
 * 6.1. Surfaces of revolution
 * 6.21. Ellipsoid
 * 6.3. Ruled surfaces
 * 6.4. Imaginary generating lines of ellipsoid, etc.
 * 6.5. Examples


 * Chap. $\text {VII}$: Elementary properties of quadric surfaces derived from their simplest equations
 * 7.1. The canonical equations
 * 7.2. Tangential properties
 * 7.3. Pole and polar
 * 7.4. Diametral planes
 * 7.5. The hyperboloids
 * 7.6. Quadric referred to conjugate diameters
 * 7.7. Normals
 * 7.8. The paraboloids
 * 7.9. Examples


 * Chap. $\text {VIII}$: The reduction of the general equation of the second degree
 * 8.1. General equation
 * 8.2. Conjugate points
 * 8.3. Invariants
 * 8.4. Polarity
 * 8.51. Canonical equations of a quadric
 * 8.6. Metrical aspect of a quadric
 * 8.7. THe discriminating cubic
 * 8.8. Transformation of rectangular coordinates
 * 8.9. Quadrics of revolution


 * Chap. $\text {IX}$: Generating lines and parametric representation
 * 9.1. Lines on a surface
 * 9.2. Equation of quadric when two generators are opposite edges of the tetrahedron of reference
 * 9.3. Regulus generated by two projective pencils of planes
 * 9.4. Lines meeting one, two, three or four fixed lines
 * 9.51. Freedom-equations of hyperboloid of one sheet
 * 9.6. Parametric equations of a curve
 * 9.7. Parametric equations of a surface
 * 9.9. Examples


 * Chap. $\text {X}$: Plane sections of a quadric
 * 10.1. Species of sections
 * 10.2. Centre of a plane section
 * 10.31. Axes of a central plane section
 * 10.4. Circular sections
 * 10.5. Models
 * 10.6. Sphere containing two circular sections
 * 10.7. Umbilics
 * 10.9. Examples


 * Chap. $\text {XI}$: Tangential equations
 * 11.1. Homogeneous point- and plane-coordinates
 * 11.21. Tangent-plane of a surface
 * 11.3. Tangential equation derived from point-equation, and vice versa
 * 11.4. Some special forms of the tangential equation of a quadric
 * 11.5. Order and class of a surface
 * 11.6. Tangential equations of a cone
 * 11.7. Equations in line-coordinates
 * 11.8. Degenerate quadric examples
 * 11.9. Examples


 * Chap. $\text {XII}$: Foci and focal properties
 * 12.1. Foci of a conic
 * 12.2. Analytical treatment
 * 12.31. Metrical property of foci
 * 12.4. Confocal quadrics
 * 12.5. The paraboloids
 * 12.61. Foci of a cone
 * 12.7. Conjugate focal conics
 * 12.8. All quadrics of a confocal system have the same foci and focal axes
 * 12.9. Deformable framework of generating lines of a quadric


 * Chap. $\text {XIII}$: Linear systems of quadrics
 * 13.1. Linear one-parameter system or pencil of quadric loci
 * 13.2. Linear tangential one-parameter system
 * 13.3. Confocal quadrics
 * 13.4. Polar properties of pencil of quadrics
 * 13.5. Polar properties of a tangential system of quadrics
 * 13.61. Quadrics through eight fixed points
 * 13.7. Paraboloids and rectangular hyperboloids in a linear system
 * 13.8. Classification of linear systems
 * 13.9. Examples


 * Chap. $\text {XIV}$: Curves and developables
 * 14.1. Curves and their representation
 * 14.21. Complex of secants and congruence of bisecants
 * 14.3. Tangents and osculating planes
 * 14.4. Developables
 * 14.51. Order, class and rank of a curve or developable
 * 14.6.
 * 14.7., two species
 * 14.71., complete intersection of two quadrics
 * 14.72.
 * 14.8. Number of intersections of two curves (conics, cubics or quartics) lying on a quadric surface
 * 14.9. Curve of striction of a regulus


 * Chap. $\text {XV}$: Invariants of a pair of quadrics
 * 15.1. Simultaneous invariants of two quadrics
 * 15.2. Geometrical meanings for the vanishing of the invariants
 * 15.31. Relation of $\Phi$ to the line-equations of the two quadrics
 * 15.4. Metrical applications
 * 15.51. Contravariants
 * 15.61. Reciprocal quadrics
 * 15.7. Harmonic complex of two quadrics
 * 15.81. Line-equation of curve of intersection of two quadrics
 * 15.91. Conjugate generators


 * Chap. $\text {XVI}$: Line geometry
 * 16.11. Plücker's coordinates
 * 16.2. Geometry of four dimensions
 * 16.3. Geometry of five dimensions
 * 16.4. Representation of lines in ordinary space by points on a $\map { {V_4}^2} \omega$; notation
 * 16.5. The linear complex
 * 16.6. Polar properties of a linear complex
 * 16.7. Canonical equation of a quadric in $S_5$
 * 16.8. The quadratic complex
 * 16.9. Special types of quadratic complexes
 * 16.95. Examples


 * Chap. $\text {XVII}$: Algebraic surfaces
 * 17.1. Definition, reducible and irreducible surfaces
 * 17.2. Curvature
 * 17.3. Polars
 * 17.4. Constant-number of an algebraic surface
 * 17.5. Double-points
 * 17.6. Lines and conics lying on a surface
 * 17.7.
 * 17.8.
 * 17.9.
 * 17.99 Examples


 * Index



Source work progress
* : Chapter $\text I$: Cartesian Coordinate-system: $1.1$. Cartesian coordinates