Author | Coxeter, H. S. M. author |
---|---|
Title | The Real Projective Plane [electronic resource] / by H. S. M. Coxeter, George Beck |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1993 |
Edition | Third Edition |
Connect to | http://dx.doi.org/10.1007/978-1-4612-2734-2 |
Descript | XIV, 227 p. online resource |
1. A Comparison of Various Kinds of Geometry -- 1ยท1 Introduction -- 1ยท2 Parallel projection -- 1ยท3 Central projection -- 1ยท4 The line at infinity -- 1ยท5 Desarguesโs two-triangle theorem -- 1ยท6 The directed angle, or cross -- 1ยท7 Hexagramma mysticum -- 1ยท8 An outline of subsequent work -- 2. Incidence -- 1ยท1 Primitive concepts -- 2ยท2 The axioms of incidence -- 2ยท3 The principle of duality -- 2ยท4 Quadrangle and quadrilateral -- 2ยท5 Harmonic conjugacy -- 2ยท6 Ranges and pencils -- 2ยท7 Perspectivity -- 2ยท8 The invariance and symmetry of the harmonic relation -- 3. Order and Continuity -- 3ยท1 The axioms of order -- 3ยท2 Segment and interval -- 3ยท3 Sense -- 3ยท4 Ordered correspondence -- 3ยท5 Continuity -- 3ยท6 Invariant points -- 3ยท7 Order in a pencil -- 3ยท8 The four regions determined by a triangle -- 4. One-Dimensional Projectivities -- 4ยท1 Projectivity -- 4ยท2 The fundamental theorem of projective geometry -- 4ยท3 Pappusโs theorem -- 4ยท4 Classification of projectivities -- 4ยท5 Periodic projectivities -- 4ยท6 Involution -- 4ยท7 Quadrangular set of six points -- 4ยท8 Projective pencils -- 5. Two-Dimensional Projectivities -- 5ยท1 Collineation -- 5ยท2 Perspective collineation -- 5ยท3 Involutory collineation -- 5ยท4 Correlation -- 5ยท5 Polarity -- 5ยท6 Polar and self-polar triangles -- 5ยท7 The self-polarity of the Desargues configuration -- 5ยท8 Pencil and range of polarities -- 5ยท9 Degenerate polarities -- 6. Conics -- 6ยท1 Historial remarks -- 6ยท2 Elliptic and hyperbolic polarities -- 6ยท3 How a hyperbolic polarity determines a conic -- 6ยท4 Conjugate points and conjugate lines -- 6ยท5 Two possible definitions for a conic -- 6ยท6 Construction for the conic through five given points -- 6ยท7 Two triangles inscribed in a conic -- 6ยท8 Pencils of conics -- 7. Projectivities on a Conic -- 7ยท1 Generalized perspectivity -- 7ยท2 Pascal and Brianchon -- 7ยท3 Construction for a projectivity on a conic -- 7ยท4 Construction for the invariant points of a given hyperbolic projectivity -- 7ยท5 Involution on a conic -- 7ยท6 A generalization of Steinerโs construction -- 7ยท7 Trilinear polarity -- 8. Affine Geometry -- 8ยท1 Parallelism -- 8ยท2 Intermediacy -- 8ยท3 Congruence -- 8ยท4 Distance -- 8ยท5 Translation and dilatation -- 8ยท6 Area -- 8ยท7 Classification of conics -- 8ยท8 Conjugate diameters -- 8ยท9 Asymptotes -- 8ยท10 Affine transformations and the Erlangen programme -- 9. Euclidean Geometry -- 9ยท1 Perpendicularity -- 9ยท2 Circles -- 9ยท3 Axes of a conic -- 9ยท4 Congruent segments -- 9ยท5 Congruent angles -- 9ยท6 Congruent transformations -- 9ยท7 Foci -- 9ยท8 Directrices -- 10. Continuity -- 10ยท1 An improved axiom of continuity -- 10ยท2 Proving Archimedesโ axiom -- 10ยท3 Proving the line to be perfect -- 10ยท4 The fundamental theorem of projective geometry -- 10ยท5 Proving Dedekindโs axiom -- 10ยท6 Enriquesโs theorem -- 11. The Introduction of Coordinates -- 11ยท1 Addition of points -- 11ยท2 Multiplication of points -- 11ยท3 Rational points -- 11ยท4 Projectivities -- 11ยท5 The one-dimensional continuum -- 11ยท6 Homogeneous coordinates -- 11ยท7 Proof that a line has a linear equation -- 11ยท8 Line coordinates -- 12. The Use of Coordinates -- 12ยท1 Consistency and categoricalness -- 12ยท2 Analytic geometry -- 12ยท3 Verifying the axioms of incidence -- 12ยท4 Verifying the axioms of order and continuity -- 12ยท5 The general collineation -- 12ยท6 The general polarity -- 12ยท7 Conies -- 12ยท8 The affine plane: affine and areal coordinates -- 12ยท9 The Euclidean plane: Cartesian and trilinear coordinates