The Real Projective Plane - Rilegato

Coxeter, H.S.M.

Rilegato

ISBN 10: 0387978895 ISBN 13: 9780387978895

Casa editrice: Springer-Verlag New York Inc., 1992

Riassunto

This introduction to projective geometry can be understood by anyone familiar with high-school geometry and algebra. The restriction to real geometry of two dimensions allows every theorem to be illustrated by a diagram. The subject is, in a sense, even simpler than Euclid, whose constructions involved a ruler and compass: here we have constructions using rulers alone. A strict axiomatic treatment is followed only to the point of letting the student see how it is done, but then relaxed to avoid becoming tedious. After two introductory chapters, the concept of continuity is introduced by means of an unusual but intuitively acceptable axiom. Subsequent chapters then treat one- and two-dimensional projectivities, conics, affine geometry, and Euclidean geometry. Chapter 10 continues the discussion of continuity at a more sophisticated level, and the remaining chapters introduce coordinates and their uses. An appendix by George Beck describes Mathematica scripts that can generate illustrations for several chapters; they are provided on a diskette included with the book. (Both PC and Macintosh versions are available) Mathematica is a registered trademark.

Su questo libro

Contenuti

1. A Comparison of Various Kinds of Geometry.- 1·1 Introduction.- 1·2 Parallel projection.- 1·3 Central projection.- 1·4 The line at infinity.- 1·5 Desargues’s two-triangle theorem.- 1·6 The directed angle, or cross.- 1·7 Hexagramma mysticum.- 1·8 An outline of subsequent work.- 2. Incidence.- 1·1 Primitive concepts.- 2·2 The axioms of incidence.- 2·3 The principle of duality.- 2·4 Quadrangle and quadrilateral.- 2·5 Harmonic conjugacy.- 2·6 Ranges and pencils.- 2·7 Perspectivity.- 2·8 The invariance and symmetry of the harmonic relation.- 3. Order and Continuity.- 3·1 The axioms of order.- 3·2 Segment and interval.- 3·3 Sense.- 3·4 Ordered correspondence.- 3·5 Continuity.- 3·6 Invariant points.- 3·7 Order in a pencil.- 3·8 The four regions determined by a triangle.- 4. One-Dimensional Projectivities.- 4·1 Projectivity.- 4·2 The fundamental theorem of projective geometry.- 4·3 Pappus’s theorem.- 4·4 Classification of projectivities.- 4·5 Periodic projectivities.- 4·6 Involution.- 4·7 Quadrangular set of six points.- 4·8 Projective pencils.- 5. Two-Dimensional Projectivities.- 5·1 Collineation.- 5·2 Perspective collineation.- 5·3 Involutory collineation.- 5·4 Correlation.- 5·5 Polarity.- 5·6 Polar and self-polar triangles.- 5·7 The self-polarity of the Desargues configuration.- 5·8 Pencil and range of polarities.- 5·9 Degenerate polarities.- 6. Conics.- 6·1 Historial remarks.- 6·2 Elliptic and hyperbolic polarities.- 6·3 How a hyperbolic polarity determines a conic.- 6·4 Conjugate points and conjugate lines.- 6·5 Two possible definitions for a conic.- 6·6 Construction for the conic through five given points.- 6·7 Two triangles inscribed in a conic.- 6·8 Pencils of conics.- 7. Projectivities on a Conic.- 7·1 Generalized perspectivity.- 7·2 Pascal and Brianchon.- 7·3 Construction for a projectivity on a conic.- 7·4 Construction for the invariant points of a given hyperbolic projectivity.- 7·5 Involution on a conic.- 7·6 A generalization of Steiner’s construction.- 7·7 Trilinear polarity.- 8. Affine Geometry.- 8·1 Parallelism.- 8·2 Intermediacy.- 8·3 Congruence.- 8·4 Distance.- 8·5 Translation and dilatation.- 8·6 Area.- 8·7 Classification of conics.- 8·8 Conjugate diameters.- 8·9 Asymptotes.- 8·10 Affine transformations and the Erlangen programme.- 9. Euclidean Geometry.- 9·1 Perpendicularity.- 9·2 Circles.- 9·3 Axes of a conic.- 9·4 Congruent segments.- 9·5 Congruent angles.- 9·6 Congruent transformations.- 9·7 Foci.- 9·8 Directrices.- 10. Continuity.- 10·1 An improved axiom of continuity.- 10·2 Proving Archimedes’ axiom.- 10·3 Proving the line to be perfect.- 10·4 The fundamental theorem of projective geometry.- 10·5 Proving Dedekind’s axiom.- 10·6 Enriques’s theorem.- 11. The Introduction of Coordinates.- 11·1 Addition of points.- 11·2 Multiplication of points.- 11·3 Rational points.- 11·4 Projectivities.- 11·5 The one-dimensional continuum.- 11·6 Homogeneous coordinates.- 11·7 Proof that a line has a linear equation.- 11·8 Line coordinates.- 12. The Use of Coordinates.- 12·1 Consistency and categoricalness.- 12·2 Analytic geometry.- 12·3 Verifying the axioms of incidence.- 12·4 Verifying the axioms of order and continuity.- 12·5 The general collineation.- 12·6 The general polarity.- 12·7 Conies.- 12·8 The affine plane: affine and areal coordinates.- 12·9 The Euclidean plane: Cartesian and trilinear coordinates.

Dettagli dell'edizione

• Editore: Springer-Verlag New York Inc.
• Data di pubblicazione: 1992
• Lingua: Inglese
• ISBN 10: 0387978895
• ISBN 13: 9780387978895
• Rilegatura: Copertina rigida
• Numero edizione: 3
• Numero di pagine: 241
• Contatto del produttore: Springer Nature Customer Service Center GmbH; ProductSafety@springernature.com; ; Europaplatz 3,69115 Heidelberg, Germany; Heidelberg; 69115; Germania