MATE-1407 Analytical Geometry

The main objective of this course is for students to expand their imagination and exercise their logic by analyzing different geometries from the axiomatic standpoint. Preliminary Notions: Euclidean and Cartesian geometry, affine planes, projective plane, duality, theorems of Desargues and Pappus, finite planes. Affine Planes: Addition and multiplication on lines, properties of operations, reciprocal of Desargues’ theorem. Co-ordinate affine planes on D division rings: coordinates, linear equations. Projective planes: Projective points, homogeneous 3D equations. Co-ordinate projective planes: Coordination, projective conics, Pascal’s theorem. Affine Space: Axiomatization, sub-geometries of an affine space, closure operator, Desargues’ Theorem, coordination. Projective Space: Axiomatization, planes in a projective space, dimension, consequences of Desargues´ theorem, coordination. Reticules of sub-geometries: Closure spaces, reticule properties. Collineations: Automorphism of planes, perspectivities in projective spaces. Fundamental theorem of projective geometry. Comparison with other non-Euclidean geometries: Spherical geometry, neutral geometry, hyperbolic geometry.