Last update, Dec. 17, 2001

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General (classics) references on Geometry:

• Coolidge, J.L. :
  • A Treatise on the Circle and the Sphere, 1916.
• Hilbert & Cohn-Vossen :
  • Geometry and the Imagination, 1932.
• Salmon, George :
  • A treatise on the analytic geometry of three dimensions, Vol.I, 7th ed., 1958.
  • A treatise on the analytic geometry of three dimensions, Vol.II, 5th ed., 1914+.

BibTeX references .

1. The Simplest Curves and Surfaces (p.1)
   1. Plane Curves
   2. The Cylinder, the Cone, the Conic Sections and Their Surfaces of Revolution
   3. The Second-Order Surfaces
   4. The Thread Construction of the Ellipsoid, and Confocal Quadrics
   5. App. 1: The Pedal-Point Construction of the Conics
   6. App. 2: The Directrices of the Conics
   7. App. 3: The Movable Rod Model of the Hyperboloid
2. Regular Systems of Points (p.32)
   1. Plane Lattices
   2. Plane Lattices in the Theory of Numbers
   3. Lattices in Three and More than Three Dimensions
   4. Crystals as Regular Systems of Points
   5. Regular Systems of Points and Discontinuous Groups of Motions
   6. Plane Motions and their Composition; Classification of the Discontinuous Groups of Motions in the Plane
   7. The Discontinuous Groups of Plane Motions with Infinite Unite Cells
   8. The Crystallographic Groups of Motions in the Plane. Regular Systems of Points and Pointers. Division of the Plane into Congruent Cells
   9. Crystallographic Classes and Groups of Motions in Space Groups and Systems of Points with Bilateral Symmetry
   10. The Regular Polyhedra
3. Projective Configurations (p.95)
   1. Preliminary Remarks about Plane Configurations
   2. The Configurations (7_3) and (8_3)
   3. The Configurations (9_3)
   4. Perspective, Ideal Elements, and the Principle of Duality in the Plane
   5. Ideal Elements and the Principle of Duality in Space. Desargues' Theorem and the Desargues Configuration (10_3)
   6. Comparison of Pascal's and Desargues Theorems
   7. Preliminary Remarks on Configurations in Space
   8. Reye's Configuration
   9. Regular Polyhedra in Three and Four Dimensions, and their Projections
   10. Enumerative Methods of Geometry
   11. Schiafli's Double-Six
4. Differential Geometry (p.172)
   1. Plane Curves
   2. Space Curves
   3. Curvature of Surfaces, Elliptic, Hyperbolic, and Parabolic Points. Lines of Curvature and Asymptotic Lines. Umbilical Points, Minimal Surfaces, Monkey Saddles
   4. The Spherical Image and Gaussian Curvature
   5. Developable Surfaces, Ruled Surfaces
   6. The Twisting of Space Curves
   7. Eleven Properties of the Sphere
   8. Bendings Leaving a Surface Invariant
   9. Elliptic Geometry
   10. Hyperbolic Geometry, and its Relation to Euclidean and to Elliptic Geometry
   11. Stereographic Projection and Circle-Preserving Transformations. Poincare's Model of the Hyperbolic Plane
   12. Methods of Mapping, Isometric, Area-Preserving, Geodesic, Continuous and Conformal Mappings
   13. Geometrical Function Theory. Riemann's Mapping Theorem. Conformal Mapping in Space
   14. Conformal Mappings of Curved Surfaces. Minimal Surfaces. Plateau's Problem
5. Kinematics (p.272)
   1. Linkages
   2. Continuous Rigid Motions of Plane Figures
   3. An Instrument for Constructing the Ellipse and its Roulettes
   4. Continuous Motions in Space
6. Topology (p.290)
   1. Polyhedra
   2. Surfaces
   3. One-Sided Surfaces
   4. The Projective Plane as a Closed Surface
   5. Topological Mappings of a Surface onto Itself. Fixed Points. Classes of Mappings. The Universal Covering Surface of the Torus
   6. Conformal Mapping of the Torus
   7. The Problem of Contiguous Regions, The Thread Problem, and the Color Problem
   8. The Projective Plane in Four-Dimensional Space
   9. The Euclidean Plane in Four-Dimensional Space
7. Index (p. 345)

Ch. 1 - Coordinates

Ch. 2 - Interpretation of (algebraic) Equations

• Every plane section of a surface of the order n is a curve of the order n. (p.15)
• Every right line meets a surface of order n in n points (some of which may be at infinity).
• 2 equations of degrees m & n respectively represent a curve of the order m·n (p.16).
  If the inverse is true, the curve is said to be a complete intersection.
• 3 equations of the degree m, n & p in general denote m·n·p points, which follows from the theory of elimination.
  3 surfaces of order m, n & p in general intersect in m·n·p points, except when they all pass through the same curve.

Ch. 3 - The Plane & the Right Line

The Plane

• Every equation of the 1st degree represent a plane and conversely.
• Pencil : Let L & M denote two planes, then the linear combination a L + b M denotes a plane passing through the line of intersection of L & M. This pencil of planes is called "büschel" in German.
• Sheaf : Let L, M & N denote three planes, then the linear combination a L + b M + c N denotes a plane passing through the point common to all three. This sheaf is called "bündel" in German.
• Special cases: a L + b denotes a plane parallel to L, and a L + b M + c denotes a plane parallel to the intersection of L and M.
• 4 planes, L, M, N & P will pass through the same point if their equations are connected by the linear relation: a L + b M + c N + d P = 0  (p.25).
• Quadri-planar coordinates : The equation of any surface of the degree n can be expressed, in general, by a homogeneous equation of the degree n between the planes, L, M, N & P.
  E.g., consider the 4 planes: x=0, y=0, z=0, w=0, then the quadri-planar coordinates (x,y,z,w) of the surface under consideration are given by four quantities proportional to the (perpendicular) distances of a surface point to each of the 4 planes. N.B: These 4 planes determine a tetrahedron, and the quadri-planar coordinates are alike the barycentric coordinates concept.

The Right Line

• A right line can be represented as the intersection of 2 planes (p.30).
  By eliminating x & y alternately between the 2 equations, we reduce them to the common form: x = m z + a, y = n z + b . Hence, the equations of a right line include 4 independent variables.

The Six Coordinates of a Right Line

(pp. 39-46)

Some Properties of Tetrahedra

• Volume of a tetrahedron in terms of its six edges (p.47).
• Radius of the circumscribing sphere to a tetrahedron (p.49).

Ch. 4 - Properties Common to all Surfaces of the 2nd Degree: Quadrics (or Conicoids)

(pp. 51-74)

General equation:

or

which has 9 independent terms, and hence, 9 points are sufficient to specify a quadric in general.

We can express the above as a homogeneous function, via the equations of 4 given planes x,y,z,w :

or

Ch. 5 - Classification of Quadrics

Let D = abc + 2fgh - af² - bg² -ch² .

Central quadrics: non-vanishing D

These quadrics have a unique centre at a finite distance from the origin.

By parallel translation of axes w/r to the centre, the linear terms can be eliminated, i.e.: l = m = n =0 .

By rotation of the axes w/r to the new origin, the mixed terms can also be eliminated, i.e.: f' = g' = h' = 0 .

Thus, central quadrics can always be written in the compact form: a'x² + b'y² + c'z² + d' = 0 .

Non-central quadrics: D = 0

Ch. 6 - Properties of Quadrics deduced from Special Forms of their Equations

Central Surfaces

Non-Central Surfaces

Surfaces of Revolution

Loci

Ch. 7 - Reciprocation, Duality, Abridged Notation & Projection

Ch. 8 - Foci & Confocal Surfaces

Ch. 9 - Invariants & Covariants of Systems of Quadrics

Ch. 10 - Cones & Sphero-Conics

Ch. 11 - General Theory of Surfaces, Curvature

Ch. 12 - Curves & Developables

Ch. 13 - PDE of Families of Surfaces

Equations involving 2 parameters & 1 arbitrary function

Cylindrical surfaces

Conical surfaces or cones

Conoidal surfaces, the right conoid, the cylindroid

Ch. 14 - The Wave Surface, the Centro-Surface, Parallel, Pedal, & Inverse Surfaces

Ch. 15 - Surfaces of the Third Degree

Ch. 16 - Surfaces of the Fourth Degree

Ch. 17 - General Theory of Surfaces

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