This volume serves as an extension of high school-level studies of geometry and algebra, and proceeds to more advanced topics with an axiomatic approach. Includes an introductory chapter on projective geometry, then explores the relations between the basic theorems; higher-dimensional space; conics; coordinate systems and linear transformations; quadric surfaces; and the Jordan canonical form. 1962 edition.
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I. Projective Geometry as an Extension of High School Geometry 1. Two approaches to projective geometry 2. An initial question 3. Projective invariants 4. Vanishing points 5. Vanishing lines 6. Some projective noninvariants 7. Betweenness 8. Division of a segment in a ratio 9. Desargues' Theorem 10. Perspectivity; projectivity 11. Harmonic tetrads; fourth harmonic 12. Further theorems on harmonic tetrads 13. The cross-ratio 14. Fundamental Theorem of Projective Geometry 15. Further remarks on the cross-ratio 16. Construction of the projective plane 17. Previous results in the constructed plane 18. Analytic construction of the projective plane 19. Elements of linear equations II. The Axiomatic Foundation 1. Unproved propositions and undefined terms 2. Requirements on the axioms and undefined terms 3. Undefined terms and axioms for a projective plane 4. Initial development of the system; the Principle of Duality 5. Consistency of the axioms 6. Other models 7. Independence of the axioms 8. Isomorphism 9. Further axioms 10. Consequences of Desargues' Theorem 11. Free planes III. Establishing Coordinates in a Plane 1. Definition of a field 2. Consistency of the field axioms 3. The analytic model 4. Geometric description of the operations plus and times 5. Setting up coordinates in the projective plane 6. The noncommutative case IV. Relations between the Basic Theorems V. Axiomatic Introduction of Higher-Dimensional Space 1. Higher-dimensional, especially 3-dimensional projective space 2. Desarguesian planes and higher-dimensional space VI. Conics 1. Study of the conic on the basis of high school geometry 2. The conic, axiomatically treated 3. The polar 4. The polar, axiomatically treated 5. Polarities VII. Higher-Dimensional Spaces Resumed 1. Theory of dependence 2. Application of the dependency theory to geometry 3. Hyperplanes 4. The dual space 5. The analytic case VIII. Coordinate Systems and Linear Transformations 1. Coordinate systems 2. Determinants 3. Coordinate systems resumed 4. Coordinate changes, alias linear transformations 5. A generalization from n = 2 to n = 1 6. Linear transformations on a line and from one line to another 7. Cross-ratio 8. Coordinate systems and linear transformations in higher-dimensional spaces 9. Coordinates in affine space IX. Coordinate Systems Abstractly Considered 1. Definition of a coordinate system 2. Definition of a geometric object 3. Algebraic curves 4. A short cut to PNK 5. A result for the field of real numbers X. Conic Sections Analytically Treated 1. Derivation of equation of conic 2. Uniqueness of the equation 3. Projective equivalence of conics 4. Poles and polars 5. Polarities and conics Appendix to Chapter X A1. Factorization of linear transformation into polarities XI. Coordinates on a conic 1. Coordinates on a conic 2. Projectivities on a conic XII. Pairs of Conics 1. Pencils of conics 2. Intersection multiplicities XIII. Quadric Surfaces 1. Projectivities between pencils of planes 2. Reguli and quadric surfaces 3. Quadric surfaces over the complex field 4. Some properties of the sphere XIV. The Jordan Canonical Form Bibliographical Note Index
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