In 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, repectively. 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 shows the connections among projective, Euclidean, and analytic geometry.
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Triangles and Quadrangles
The Principle of Duality
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analytic geometry axial pencil Axiom axis bundle of parallels coincide collinear points common point complete quadrangle concurrent lines conic conjugate lines conjugate points construction coordinates coplanar corresponding points deﬁned deﬁnition Desargues determined diagonal points diagonal triangle dual Dualize Dually elation elliptic equation Euclidean geometry EXERCISES exterior point ﬁeld ﬁgure Figure 2.3A ﬁnd ﬁnite ﬁrst ﬁve points ﬁxed point given points H(AB harmonic conjugate harmonic homology harmonic set Hence hexagon hyperbolic polarity incident inscribed interchanges intersection invariant line invariant point involution join line at inﬁnity locus meet number of points parabolic projectivity parallel planes pass perspective collineation point at inﬁnity points and lines projective collineation projective correlation projective geometry projective plane PROOF quadrangle PQRS quadrilateral S. L. Greitzer secant Section self-conjugate point self-polar triangle sides tangents three collinear points three diagonal three lines transforms triangle of reference triangle PQR trilinear pole vertex vertices