Projective geometry and modern algebra

Description

An introductory course on projective geometry. The authors develop the elementary synthetic theory of projective planes, along with an analytic discussion of the real projective plane and examples that show the independence of the axioms. After a motivating historical foreword, the main body of the work consists of the following chapters: 1. Affine geometry. 2. Projective planes. 3. Desargues' theorem and the principle of duality. 4. A brief introduction to groups. 5. Elementary synthetic projective geometry. 6. The fundamental theorem for projectivities on a line. 7. A brief introduction to division rings. 8. Projective planes over division rings. 9. Introduction of coordinates in a projective plane. 10. Möbius transformations and cross ratio. 11. Projective collineations. Along with the geometry, all algebraic concepts that are used are introduced properly and furnished with examples. It may be worth mentioning that apart from the usual example for a division ring (quaternions) the authors also discuss skew Laurent series in Chapter 7. After the main body there are five appendices, intended to introduce the student to further reading and independent work: A. Conics. B. Algebraic curves and Bezout's theorem. C. Elliptic geometry. D. Ternary rings. E. The lattice of subspaces.