Fundamental Concepts of Geometry (eBook)

CHAPTER 1. FOUNDATIONS OF GEOMETRY1-1 Logical systems1-2 Logical notations1-3 Inductive and deductive reasoning1-4 Postulates1-5 Independent postulates1-6 Categorical sets of postulates1-7 A geometry of number triples1-8 Geometric invariantsCHAPTER 2. SYNTHETIC PROJECTIVE GEOMETRY2-1 Postulates of incidence and existence2-2 Properties of a projective plane2-3 Figures2-4 Duality2-5 Perspective figures2-6 Projective transformations2-7 Postulate of Projectivity2-8 Quadrangles2-9 Complete and simple n-points2-10 Theorem of Desargues2-11 Theorem of Pappus2-12 Conics2-13 Theorem of Pascal2-14 SurveyCHAPTER 3. COORDINATE SYSTEMS3-1 Quadrangular sets3-2 Properties of quadrangular sets3-3 Harmonic sets3-4 Postulates of Separation3-5 Nets of rationality3-6 Real projective geometry3-7 Nonhomogeneous coordinates3-8 Homogeneous coordinates3-9 SurveyCHAPTER 4. ANALYTIC PROJECTIVE GEOMETRY4-1 Representations in space4-2 Representations on a plane4-3 Representations on a line4-4 Matrices4-5 Cross ratio4-6 Analytic and synthetic geometries4-7 Groups4-8 Classification of projective transformations4-9 Polarities and conics4-10 Conics4-11 Involutions on a line4-12 SurveyCHAPTER 5. AFFINE GEOMETRY5-1 Ideal points5-2 Parallels5-3 Mid-point5-4 Classification of conics5-5 Affine transformations5-6 Homothetic transformations5-7 Translations5-8 Dilations5-9 Line reflections5-10 Equiaffine and equiareal transformations5-11 SurveyCHAPTER 6. EUCLIDEAN PLANE GEOMETRY6-1 Perpendicluar lines6-2 Similarity transformations6-3 Orthogonal line reflections6-4 Euclidean transformations6-5 Distances6-6 Directed angles6-7 Angles6-8 Common figures6-9 SurveyCHAPTER 7. THE EVOLUTION OF GEOMETRY7-1 Early measurements7-2 Early Greek influence7-3 Euclid7-4 Early euclidean geometry7-5 The awakening in Europe7-6 Constructions7-7 Descriptive geometry7-8 Seventeenth century7-9 Eighteenth century7-10 Euclid's fifth postulate7-11 Nineteenth and twentieth centuries7-12 SurveyCHAPTER 8. NONEUCLIDEAN GEOMETRY8-1 The absolute polarity8-2 Points and lines8-3 Hyperbolic geometry8-4 Elliptic and spherical geometries8-5 ComparisonsCHAPTER 9. TOPOLOGY9-1 Topology9-2 Homeomorphic figures9-3 Jordan Curve Theorem9-4 Surfaces9-5 Euler's Formula9-6 Tranversable networks9-7 Four-color problem9-8 Fixed-point theorems9-9 Moebius strip9-10 SurveyBIBLIOGRAPHYINDEX