Modern Geometries

by Smart, James R.

• ISBN13: 9780534351885
• ISBN10: 0534351883
• Edition: 5th
• Format: Hardcover
• Copyright: 1997-12-15
• Publisher: Cengage Learning

Summary

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.

Table of Contents

1. Sets of Axioms and Finite Geometries	1	(36)
1.1 Introduction to Geometry	1	(6)
1.2 Development of Modern Geometries	7	(5)
1.3 Introduction to Finite Geometries	12	(5)
1.4 Four-Line and Four-Point Geometries	17	(4)
1.5 Finite Geometries of Fano and Young	21	(5)
1.6 Finite Geometries of Pappus and Desargues	26	(5)
1.7 Other Finite Geometries	31	(6)
2. Geometric Transformations	37	(72)
2.1 Introduction to Transformations	37	(7)
2.2 Groups of Transformations	44	(7)
2.3 Euclidean Motions of the Plane	51	(10)
2.4 Sets of Equations for Motions of the Plane	61	(8)
2.5 Applications of Transformations in Computer Graphics	69	(7)
2.6 Properties of the Group of Euclidean Motions	76	(8)
2.7 Motions and Graphics of Three-Space	84	(8)
2.8 Similarity Transformations	92	(6)
2.9 Introduction to the Geometry of Fractals and Fractal Dimension	98	(4)
2.10 Examples and Applications of Fractals	102	(7)
3. Convexity	109	(46)
3.1 Basic Concepts	109	(8)
3.2 Convex Sets and Supporting Lines	117	(7)
3.3 Convex Bodies in Two-Space	124	(7)
3.4 Convex Bodies in Three-Space	131	(5)
3.5 Convex Hulls	136	(5)
3.6 Width of a Set	141	(6)
3.7 Helly's Theorem and Applications	147	(8)
4. Modern Euclidean Geometry, Theory, and Applications	155	(56)
4.1 Fundamental Concepts and Theorem	155	(10)
4.2 Some Theorems Leading to Modern Synthetic Geometry	165	(7)
4.3 The Nine-Point Circle and Early Nineteenth-Century Synthetic Geometry	172	(5)
4.4 Isogonal Conjugates	177	(5)
4.5 Recent Synthetic Geometry of the Triangle	182	(5)
4.6 Golden Ratio, Tessellations, Packing Problems, and Pick's Theorem	187	(8)
4.7 Extremum Problems, Geometric Probability, Fuzzy Sets, and Bezier Curves	195	(16)
5. Constructions	211	(32)
5.1 The Philosophy of Constructions	211	(5)
5.2 Constructible Numbers	216	(4)
5.3 Constructions in Advanced Euclidean Geometry	220	(6)
5.4 Constructions and Impossibility Proofs	226	(8)
5.5 Constructions by Paper Folding and by Use of Computer Software	234	(3)
5.6 Constructions with Only One Instrument	237	(6)
6. The Transformation of Inversion	243	(30)
6.1 Basic Concepts	243	(6)
6.2 Additional Properties and Invariants Under Inversion	249	(6)
6.3 The Analytic Geometry of Inversion	255	(6)
6.4 Some Applications of Inversion	261	(12)
7. Projective Geometry	273	(62)
7.1 Fundamental Concepts	273	(6)
7.2 Postulation Basis for Projective Geometry	279	(3)
7.3 Duality and Some Consequences	282	(6)
7.4 Harmonic Sets	288	(6)
7.5 Projective Transformations	294	(6)
7.6 Homogeneous Coordinates	300	(8)
7.7 Equations for Projective Transformations	308	(9)
7.8 Special Projectivities	317	(5)
7.9 Conics	322	(6)
7.10 Construction of Conics	328	(7)
8. Geometric Introduction to Topological Transformations	335	(30)
8.1 Topological Transformations	335	(5)
8.2 Simple Closed Curves	340	(7)
8.3 Invariant Points and Networks	347	(4)
8.4 Introduction to the Topology of Surfaces	351	(4)
8.5 Euler's Formula and the Map-Coloring Problem	355	(10)
9. Non-Euclidean Geometries	365	(42)
9.1 Foundations of Euclidean and Non-Euclidean Geometries	365	(5)
9.2 Introduction to Hyerbolic Geometry	370	(5)
9.3 Ideal Points and Omega Triangles	375	(4)
9.4 Quadrilaterals and Triangles	379	(5)
9.5 Pairs of Lines and Area of Triangular Regions	384	(5)
9.6 Curves	389	(4)
9.7 Elliptic Geometry	393	(4)
9.8 Consistency; Other Modern Geometries	397	(10)
Appendix 1 Selected Ideas from Logic	407	(2)
Appendix 2 Review of Elementary Euclidean Geometry	409	(3)
Appendix 3 The First Twenty-eight Propositions of Euclid (Revised)	412	(2)
Appendix 4 Hilbert's Axioms	414	(3)
Appendix 5 Birkhoff's Postulates	417	(1)
Appendix 6 Illustrations of Basic Euclidean Constructions	418	(3)
Bibliography	421	(6)
Answers to Selected Exercises	427	(16)
Index	443