Projective Geometry and Modern Algebra

Historical foreword. Affine geometry: affine planes; transformations of the affine plane. Projective planes: completion of the affine plane; homogeneous coordinates for the real projective plane. Desargues' theorem and the principle of duality: the axiom P5 of Desargues; Moulton's example; axioms for projective space; principle of duality. A brief introduction to groups: elements of group theory; automorphisms of the projective plane of 7 points. Elementary synthetic projective geometry: Fano's axiom P6; harmonic points; perpectivities and projectivities. The fundamental theorem for projectivities on a line: the fundamental theorem: axiom P7; geometry of complex numbers; Pappu's theorem. A brief introduction to division rings: division rings; the quaternions H; a noncommutative division ring with characteristics p. Projective planes over division rings: P2(R); the automorphism group of P2 (R); the algebraic meaning of axioms P6 and P7; independence of axioms. Introduction of coordinates in a projective plane: the major and minor Desargues' axioms; division ring number lines; introducing coordinates in A. Mobius transformations and cross ratio: assessment; Mobius transformations of the extended field; cross ratio: a projective invariant. Projective collineations: projective collineations; elations and homologies; the fundamental theorem of projective collineation; Ceva's theorem. (Part contents).