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

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## D.M.Y. Sommerville:

## D.M.Y. Sommerville: *Analytical Geometry of Three Dimensions*

Published $\text {1934}$, **Cambridge at the University Press**.

### Subject Matter

### 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

- 6.1. Surfaces of revolution

*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

- 11.1. Homogeneous point- and plane-coordinates

*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. The Space Cubic
- 14.7. Quartic Curves, two species
- 14.71. Quartics of First Species, complete intersection of two quadrics
- 14.72. Quartics of Second Species

- 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

- 14.1. Curves and their representation

*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. Ruled Surfaces
- 17.8. Cubic Surfaces
- 17.9. Quartic Surfaces
- 17.99 Examples

*Index*

## Source work progress

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