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Author Coxeter, H. S. M. author The Real Projective Plane [electronic resource] / by H. S. M. Coxeter, George Beck New York, NY : Springer New York : Imprint: Springer, 1993 Third Edition http://dx.doi.org/10.1007/978-1-4612-2734-2 XIV, 227 p. online resource

SUMMARY

Along with many small improvements, this revised edition contains van Yzeren's new proof of Pascal's theorem (ยง1.7) and, in Chapter 2, an improved treatment of order and sense. The Sylvester-Gallai theorem, instead of being introduced as a curiosity, is now used as an essential step in the theory of harmonic separation (ยง3.34). This makes the logiยญ cal development self-contained: the footnotes involving the References (pp. 214-216) are for comparison with earlier treatments, and to give credit where it is due, not to fill gaps in the argument. H.S.M.C. November 1992 v Preface to the Second Edition Why should one study the real plane? To this question, put by those who advocate the complex plane, or geometry over a general field, I would reply that the real plane is an easy first step. Most of the propยญ erties are closely analogous, and the real field has the advantage of intuitive accessibility. Moreover, real geometry is exactly what is needed for the projective approach to nonยท Euclidean geometry. Instead of introducing the affine and Euclidean metrics as in Chapters 8 and 9, we could just as well take the locus of 'points at infinity' to be a conic, or replace the absolute involution by an absolute polarity

CONTENT

1. A Comparison of Various Kinds of Geometry -- 1ยท1 Introduction -- 1ยท2 Parallel projection -- 1ยท3 Central projection -- 1ยท4 The line at infinity -- 1ยท5 Desarguesโ{128}{153}s 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 Pappusโ{128}{153}s 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 Steinerโ{128}{153}s 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{128}{153} axiom -- 10ยท3 Proving the line to be perfect -- 10ยท4 The fundamental theorem of projective geometry -- 10ยท5 Proving Dedekindโ{128}{153}s axiom -- 10ยท6 Enriquesโ{128}{153}s 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

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