Introduction to Non-Euclidean Geometry (eBook)

Eisenreich (Autor)

eBook Download: PDF

2014 | 1. Auflage
274 Seiten
Elsevier Science (Verlag)
978-1-4832-9531-2 (ISBN)

Infos

An Introduction to Non-Euclidean Geometry covers some introductory topics related to non-Euclidian geometry, including hyperbolic and elliptic geometries. This book is organized into three parts encompassing eight chapters. The first part provides mathematical proofs of Euclid's fifth postulate concerning the extent of a straight line and the theory of parallels. The second part describes some problems in hyperbolic geometry, such as cases of parallels with and without a common perpendicular. This part also deals with horocycles and triangle relations. The third part examines single and double elliptic geometries. This book will be of great value to mathematics, liberal arts, and philosophy major students.

Inhalt

Front Cover 1
An Introduction to Non-Euclidean Geometry 4
Copyright Page 5
Table of Contents 8
Dedication 6
Preface 12
PART I: HISTORICAL INTRODUCTION 14
Chapter I. Euclid's Fifth Postulate 16
1. Introduction 16
2. Euclid's Stated Assumptions 17
3. The Extent of a Straight Line 18
4. Euclid's Theory of Parallels 20
5. Further Consequences of Postulate 5 23
6. Substitutes for Postulate 5 25
Chapter II. Attempts to Prove the Fifth Postulate 29
1. Introduction 29
2. Euclid's Choices 29
3. Posidonius and His Followers 31
4. Ptolemy and Proclus 32
5. Saccheri 35
6. Lambert 39
7. Legendre 40
8. The Discovery of Non-Euclidean Geometry 43
PART II: HYPERBOLIC GEOMETRY 46
Chapter III. Parallels With a Common Perpendicular 48
1. Introduction 48
2. The Basis E 48
3. The Initial Theorems of Hyperbolic Geometry 52
4. The Hyperbolic Parallel Postulate 57
5. Immediate Consequences of the Postulate 58
6. Further Properties of Quadrilaterals 60
7. Parallels With a Common Perpendicular 62
8. The Angle-Sum of a Triangle 68
9. The Defect of a Triangle 72
10. Quadrilaterals Associated with a Triangle 76
11. The Equivalence of Triangles 78
12. Area of a Triangle 82
13. Implications of the Area Formula 85
14. Circles 89
Chapter IV. Parallels Without a Common Perpendicular 94
1. Introduction 94
2. Parallels Without a Common Perpendicular 94
3. Properties of Boundary Parallels 101
4. Trilateral 108
5. Angles of Parallelism 113
6. Distance between Two Lines 119
7. The Uniqueness of Parallels Without a Common Perpendicular 124
8. Perpendicular Bisectors of the Sides of a Triangle 126
Chapter V. Horocycles 129
1. Introduction 129
2. Corresponding Points 129
3. Definition of a Horocycle 134
4. Arcs and Chords of a Horocycle 136
5. Codirectional Horocycles 144
6. Arc Length on a Horocycle 154
7. Formulas Related to k-Arcs 158
Chapter VI. Triangle Relations 164
1. Introduction 164
2. Associated Right Triangles 164
3. Improved Angle of Parallelism Formulas 170
4. Remarks on the Trigonometric Functions 173
5. Right Triangle Formulas 175
6. Comparison with Euclidean Formulas 181
7. Formulas for the General Triangle 188
8. Hyperbolic Geometry in Small Regions 193
9. Hyperbolic Geometry and the Physical World 197
PART III: ELLIPTIC GEOMETRY 204
Chapter VII. Double Elliptic Geometry 206
1. Introduction 206
2. Riemann 207
3. The Elliptic Geometries 209
4. Geometry on a Sphere 210
5. A Description of Double Elliptic Geometry 213
6. Double Elliptic Geometry and the Physical World 218
7. An Axiomatic Presentation of Double Elliptic Geometry 219
Chapter VIII. Single Elliptic Geometry 249
1. Intoduction 249
2. Geometry on a Modified Hemisphere 249
3. A Description of Single Elliptic Geometry 255
4. An Axiomatic Presentation of Single Elliptic Geometry 259
Appendix: A Summary from Euclid's Elements 270
Bibliography 276
Answers to Selected Exercises 278
Index 284

Produkt-Details

• Erscheint lt. Verlag: 28.6.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 1-4832-9531-1 / 1483295311
• ISBN-13: 978-1-4832-9531-2 / 9781483295312