The purpose of this book is to revive some of the beautiful results obtained by various geometers of the 19th century, and to give its readers a taste of concrete algebraic geometry. A good deal of space is devoted to cross-ratios, conics, quadrics, and various interesting curves and surfaces. The fundamentals of projective geometry are efficiently dealt with by using a modest amount of linear algebra. An axiomatic characterization of projective planes is also given. While the topology of projective spaces over real and complex fields is described, and while the geometry of the complex projective libe is applied to the study of circles and Möbius transformations, the book is not restricted to these fields. Interesting properties of projective spaces, conics, and quadrics over finite fields are also given. This book is the first volume in the Readings in Mathematics sub-series of the UTM. From the reviews: "...The book of P. Samuel thus fills a gap in the literature. It is a little jewel. Starting from a minimal background in algebra, he succeeds in 160 pages in giving a coherent exposition of all of projective geometry. ... one reads this book like a novel. " D.Lazard in Gazette des Mathématiciens#1
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OneDimensional Projective Geometry
Classification of Conics and Quadrics
3 other sections not shown
2,2)-correspondence affine plane affine space affine subspace algebraically closed assume automorphism base points bijection called characteristic circle coefficients common points concurrent lines contains Corollary correspondence cross-ratio cubic curve cuspidal cyclic points defined degenerate conics degree denote dimension distinct double point equivalent Euclidean field finite fixed points foci given homogeneous coordinates homogeneous equation homogeneous polynomial homothety hyperplane inversion involution isomorphic isotropic lines lemma line at infinity linear forms linear system lines going multiple points non-degenerate non-zero orthogonal pairs parabola paraboloid parallel parameter parametric representation pencil of lines point at infinity polar pole projective frame projective line projective linear space projective space projective transformation Proof proper conic quadratic form quadric quartic rational function rational map rational point resp respect roots scalar self-polar shows symmetric tangent tangential equation theorem triangle unicursal unique vector space zero