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

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

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

### Contents

Preface
Chap. $\text {I}$: Cartesian coordinate-system
1.1. Cartesian coordinates
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
12.5. 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
13.4. 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
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
15.7. 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

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