This comprehensive, best-selling text focuses on the study of many different geometries -- rather than a single geometry -- and is thoroughly modern in its approach. Each chapter is essentially a short course on one aspect of modern geometry, including finite geometries, the geometry of transformations, convexity, advanced Euclidian geometry, inversion, projective geometry, geometric aspects of topology, and non-Euclidean geometries. This edition reflects the recommendations of the COMAP proceedings on Geometry's Future, the NCTM standards, and the Professional Standards for Teaching Mathematics. References to a new companion text, Active Geometry by David A. Thomas encourage students to explore the geometry of motion through the use of computer software. Using Active Geometry at the beginning of various sections allows professors to give students a somewhat more intuitive introduction using current technology before moving on to more abstract concepts and theorems.
Author(s): James R. Smart
Edition: 5
Publisher: Brooks Cole
Year: 1997
Language: English
Pages: 480
Cover......Page 1
Title page......Page 2
Preface......Page 6
1.1 Introduction to Geometry......Page 18
1.2 Development of Modern Geometries......Page 24
1.3 Introduction to Finite Geometries......Page 29
1.4 Four-Line and Four-Point Geometries......Page 34
1.5 Finite Geometries of Fano and Young......Page 38
1.6 Finite Geometries of Pappus and Desargues......Page 43
1.7 Other Finite Geometries......Page 48
2.1 Introduction to Transformations......Page 54
2.2 Groups of Transformations......Page 61
2.3 Euclidean Motions of the Plane......Page 68
2.4 Sets of Equations for Motions of the Plane......Page 78
2.5 Applications of Transformations in Computer Graphics......Page 86
2.6 Properties of the Group of Euclidean Motions......Page 93
2.7 Motions and Graphics of Three-Space......Page 101
2.8 Similarity Transformations......Page 109
2.9 Introduction to the Geometry of Fractals and Fractal Dimension......Page 115
2.10 Examples and Applications of Fractals......Page 119
3.1 Basic Concepts......Page 126
3.2 Convex Sets and Supporting lines......Page 134
3.3 Convex Bodies in Two-Space......Page 141
3.4 Convex Bodies in Three-Space......Page 148
3.5 Convex Hulls......Page 153
3.6 Width of a Set......Page 158
3.7 Helly's Theorem and Applications......Page 164
4.1 Fundamental Concepts and Theorem......Page 172
4.2 Some Theorems Leading to Modern Synthetic Geometry......Page 182
4.3 The Nine-Point Circle and Early Nineteenth-Century Synthetic Geometry......Page 189
4.4 Isogonal Conjugates......Page 194
4.5 Recent Synthetic Geometry of the Triangle......Page 199
4.6 Golden Ratio, Tessellations, Packing Problems, and Pick's Theorem......Page 204
4.7 Extremum Problems, Geometric Probability, Fuzzy Sets, and Bezier Curves......Page 212
5.1 The Philosophy of Constructions......Page 228
5.2 Constructible Numbers......Page 233
5.3 Constructions in Advanced Euclidean Geometry......Page 237
5.4 Constructions and Impossibility Proofs......Page 243
5.5 Constructions by Paper Folding and by Use of Computer Software......Page 251
5.6 Constructions with Only One Instrument......Page 254
6.1 Basic Concepts......Page 260
6.2 Additional Properties and Invariants Under Inversion......Page 266
6.3 The Analytic Geometry of Inversion......Page 272
6.4 Some Applications of Inversion......Page 278
7.1 Fundamental Concepts......Page 290
7.2 Postulation Basis for Projective Geometry......Page 296
7.3 Duality and Some Consequences......Page 299
7.4 Harmonie Sets......Page 305
7.5 Projective Transformations......Page 311
7.6 Homogeneous Coordinates......Page 317
7.7 Equations for Projective Transformations......Page 325
7.8 Special Projectivities......Page 334
7.9 Conics......Page 339
7.10 Construction of Conics......Page 345
8.1 Topological Transformations......Page 352
8.2 Simple Closed Curves......Page 357
8.3 Invariant Points and Networks......Page 364
8.4 Introduction to the Topology of Surfaces......Page 368
8.5 Euler's Formula and the Map-Coloring Problem......Page 372
9.1 Foundations of Euclidean and Non-Euclidean Geometries......Page 382
9.2 Introduction to Hyperbolic Geometry......Page 387
9.3 Ideal Points and Omega Triangles......Page 392
9.4 Quadrilaterals and Triangles......Page 396
9.5 Pairs of Lines and Area of Triangular Regions......Page 401
9.6 Curves......Page 406
9.7 Elliptic Geometry......Page 410
9.8 Consistency; Other Modern Geometries......Page 414
Appendix 1 Selected Ideas from Logic......Page 424
Appendix 2 Review of Elementary Euclidean Geometry......Page 426
Appendix 3 The First Twenty-eight Propositions of Euclid (Revised)......Page 429
Appendix 4 Hilbert's Axioms......Page 431
Appendix 5 Birkhoff's Postulates......Page 434
Appendix 6 Illustrations of Basic Euclidean Constructions......Page 435
Bibliography......Page 438
Answers to Selected Exercises......Page 444
Index......Page 460
