Modern Geometries
1. SETS OF AXIOMS AND FINITE GEOMETRIES Introduction to Geometry / Development of Modern Geometries / Introduction to Finite Geometries / Four-Line and Four-Point Geometries / Finite Geometries of Fano and Young / Finite Geometries of Pappus and Desargues / Other Finite Geometries 2. GEOMETRIC TRANSFORMATIONS Introduction to Transformations / Groups of Transformations / Euclidean Motions of the Plane / Sets of Equations for Motions of the Plane / Applications of Transformations in Computer Graphics / Properties of the Group of Euclidean Motions / Motions and Graphics of Three-Space / Similarity Transformations / Introduction to the Geometry of Fractals and Fractal Dimension / Examples and Applications of Fractals 3. CONVEXITY Basic Concepts / Convex Sets and Supporting Lines / Convex Bodies in Two-Space / Convex Bodies in Three-Space / Convex Hulls / Width of a Set / Helly's Theorem and Applications 4. MODERN EUCLIDEAN GEOMETRY, THEORY, AND APPLICATIONS Fundamental Concepts and Theorems / Some Theorems Leading to Modern Synthetic Geometry / The Nine-Point Circle and Early Nineteenth-Century Synthetic Geometry / Isogonal Conjugates / Recent Synthetic Geometry of the Triangle / Golden Ratio, Tessellations, Packing Problems and Pick's Theorem / Extremum Problems, Geometric Probability, Fuzzy Sets, and Bezier Curves 5. CONSTRUCTIONS The Philosophy of Constructions / Constructible Numbers / Constructions in Advanced Euclidean Geometry / Constructions and Impossibility Proofs / Constructions by Paper Folding and by Use of Computer Software / Constructions with Only One Instrument 6. THE TRANSFORMATION OF INVERSION Basic Concepts / Additional Properties and Invariants under Inversion / The Analytic Geometry of Inversion / Some Applications of Inversion 7. PROJECTIVE GEOMETRY Fundamental Concepts / Postulational Basis for Projective Geometry / Duality and Some Consequences / Harmonic Sets / Projective Transformations / Homogenous Coordinates / Equations for Projective Transformations / Special Projectivities / Conics / Constructions of Conics 8. GEOMETRIC INTRODUCTION TO TOPOLOGICAL TRANSFORMATIONS Topological Transformations / Simple Closed Curves / Invariant Points and Networks / Introduction to the Topology of Surfaces / Euler's Formula and the Map-Coloring Problem 9. NON-EUCLIDEAN GEOMETRIES Foundations of Euclidean and Non-Euclidean Geometries / Introduction to Hyperbolic Geometry / Ideal Points and Omega Triangles / Quadrilaterals and Triangles / Pairs of Lines and Area of Triangular Regions / Curves / Elliptic Geometry / Consistency; Other Modern Geometries / APPENDIX 1: SELECTED IDEAS FROM LOGIC / APPENDIX 2: REVIEW OF EUCLIDEAN GEOMETRY / APPENDIX 3: FIRST TWENTY-EIGHT PROPOSITIONS OF EUCLID / APPENDIX 4: HILBERT'S AXIOMS / APPENDIX 5: BIRKHOFF'S POSTULATES / APPENDIX 6: ILLUSTRATIONS OF BASIC EUCLIDEAN CONSTRUCTIONS / BIBLIOGRAPHY / ANSWERS TO SELECTED EXERCISES / INDEX