Projective GeometryIn Euclidean geometry, constructions are made with ruler and compass. Projective geometry is simpler: its constructions require only a ruler. In projective geometry one never measures anything, instead, one relates one set of points to another by a projectivity. The first two chapters of this book introduce the important concepts of the subject and provide the logical foundations. The third and fourth chapters introduce the famous theorems of Desargues and Pappus. Chapters 5 and 6 make use of projectivities on a line and plane, respectively. The next three chapters develop a self-contained account of von Staudt's approach to the theory of conics. The modern approach used in that development is exploited in Chapter 10, which deals with the simplest finite geometry that is rich enough to illustrate all the theorems nontrivially. The concluding chapters show the connections among projective, Euclidean, and analytic geometry. |
Contents
Introduction | 1 |
The Fundamental Theorem and Pappuss Theorem | 33 |
Onedimensional Projectivities | 41 |
Copyright | |
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Common terms and phrases
a₁ ABC)(Pp affine geometry affine space axial pencil Axiom axis B₂ bundle of parallels C₁ Chasles's theorem coincide collinear points common point complete quadrangle concurrent lines conic conjugate lines conjugate points construction coordinates coplanar corresponding points Desargues Desargues configuration determined diagonal points diagonal triangle dual Dualize Dually elation elliptic equation Euclidean geometry exterior point Figure 2.3A given points H(AB harmonic conjugate harmonic homology harmonic set Hence hexagon hyperbolic polarity incident inscribed interchanges intersection invariant point involution join line at infinity locus meet O₁ P₁ parabolic projectivity pass perspective collineation point at infinity points and lines projective collineation projective correlation projective geometry projective plane PROOF quadrangle PQRS quadrilateral secant Section self-conjugate points self-polar triangle sides tangents three collinear points three diagonal three distinct points three lines transforms triangle of reference triangle PQR trilinear pole vertex vertices X₁