Classical Geometry ：Euclidean, Transformational, Inversive, and Projective

Publication subTitle ：Euclidean, Transformational, Inversive, and Projective

Author： I. E. Leonard  

Publisher： John Wiley & Sons Inc‎

Publication year： 2014

E-ISBN: 9781118839430

P-ISBN(Hardback):  9781118679197

Subject： O18 geometric topology

Language： ENG

Access to resources Favorite

Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.

Description

Features the classical themes of geometry with plentiful applications in mathematics, education, engineering, and science

Accessible and reader-friendly, Classical Geometry: Euclidean, Transformational, Inversive, and Projective introduces readers to a valuable discipline that is crucial to understanding bothspatial relationships and logical reasoning. Focusing on the development of geometric intuitionwhile avoiding the axiomatic method, a problem solving approach is encouraged throughout.

The book is strategically divided into three sections: Part One focuses on Euclidean geometry, which provides the foundation for the rest of the material covered throughout; Part Two discusses Euclidean transformations of the plane, as well as groups and their use in studying transformations; and Part Three covers inversive and projective geometry as natural extensions of Euclidean geometry. In addition to featuring real-world applications throughout, Classical Geometry: Euclidean, Transformational, Inversive, and Projective includes:

  • Multiple entertaining and elegant geometry problems at the end of each section for every level of study
  • Fully worked examples with exercises to facilitate comprehension and retention
  • Unique topical coverage, such as the theorems of Ceva and Menalaus and their applications
  • An approach that prepares readers for the art of logical reasoning, modeling, and proofs

The book is an excellent textbook for courses in introductory geometry, elementary geometry, modern geometry, and history of mathematics at the undergraduate level for mathematics majors, as well as for engineering and secondary education majors. The book is also ideal for anyone who would like to learn the various applications of elementary geometry.

Chapter

CLASSICAL GEOMETRY: Euclidean, Transformational, Inversive, and Projective

Copyright

CONTENTS

Preface

PART I EUCLIDEAN GEOMETRY

1 PART I EUCLIDEAN GEOMETRY Congruency

1.1 Introduction

1.2 Congruent Figures

1.3 Parallel Lines

1.3.1 Angles in a Triangle

1.3.2 Thales' Theorem

1.3.3 Quadrilaterals

1.4 More About Congruency

1.5 Perpendiculars and Angle Bisectors

1.6 Construction Problems

1.6.1 The Method of Loci

1.7 Solutions to Selected Exercises

1.8 Problems

2 Concurrency

2.1 Perpendicular Bisectors

2.2 Angle Bisectors

2.3 Altitudes

2.4 Medians

2.5 Construction Problems

2.6 Solutions to the Exercises

2.7 Problems

3 Similarity

3.1 Similar Triangles

3.2 Parallel Lines and Similarity

3.3 Other Conditions Implying Similarity

3.4 Examples

3.5 Construction Problems

3.6 The Power of a Point

3.7 Solutions to the Exercises

3.8 Problems

4 Theorems of Ceva and Menelaus

4.1 Directed Distances, Directed Ratios

4.2 The Theorems

4.3 Applications of Ceva's Theorem

4.4 Applications of Menelaus' Theorem

4.5 Proofs of the Theorems

4.6 Extended Versions of the Theorems

4.6.1 Ceva's Theorem in the Extended Plane

4.6.2 Menelaus' Theorem in the Extended Plane

4.7 Problems

5 Area

5.1 Basic Properties

5.1.1 Areas of Polygons

5.1.2 Finding the Area of Polygons

5.1.3 Areas of Other Shapes

5.2 Applications of the Basic Properties

5.3 Other Formulae for the Area of a Triangle

5.4 Solutions to the Exercises

5.5 Problems

6 Miscellaneous Topics

6.1 The Three Problems of Antiquity

6.2 Constructing Segments of Specific Lengths

6.3 Construction of Regular Polygons

6.3.1 Construction of the Regular Pentagon

6.3.2 Construction of Other Regular Polygons

6.4 Miquel's Theorem

6.5 Morley's Theorem

6.6 The Nine-Point Circle

6.6.1 Special Cases

6.7 The Steiner-Lehmus Theorem

6.8 The Circle of Apollonius

6.9 Solutions to the Exercises

6.10 Problems

PART II TRANSFORMATIONAL GEOMETRY

7 The Euclidean Transformations or lsometries

7.1 Rotations, Reflections, and Translations

7.2 Mappings and Transformations

7.2.1 Isometries

7.3 Using Rotations, Reflections, and Translations

7.4 Problems

8 The Algebra of lsometries

8.1 Basic Algebraic Properties

8.2 Groups of Isometries

8.2.1 Direct and Opposite Isometries

8.3 The Product of Reflections

8.4 Problems

9 The Product of Direct lsometries

9.1 Angles

9.2 Fixed Points

9.3 The Product of Two Translations

9.4 The Product of a Translation and a Rotation

9.5 The Product of Two Rotations

9.6 Problems

10 Symmetry and Groups

10.1 More About Groups

10.1.1 Cyclic and Dihedral Groups

10.2 Leonardo's Theorem

10.3 Problems

11 Homotheties

11.1 The Pantograph

11.2 Some Basic Properties

11.2.1 Circles

11.3 Construction Problems

11.4 Using Homotheties in Proofs

11.5 Dilatation

11.6 Problems

12 Tessellations

12.1 Tilings

12.2 Monohedral Tilings

12.3 Tiling with Regular Polygons

12.4 Platonic and Archimedean Tilings

12.5 Problems

PART Ill INVERSIVE AND PROJECTIVE GEOMETRIES

13 Introduction to Inversive Geometry

13.1 Inversion in the Euclidean Plane

13.2 The Effect of Inversion on Euclidean Properties

13.3 Orthogonal Circles

13.4 Compass-Only Constructions

13.5 Problems

14 Reciprocation and the Extended Plane

14.1 Harmonic Conjugates

14.2 The Projective Plane and Reciprocation

14.3 Conjugate Points and Lines

14.4 Conics

14.5 Problems

15 Cross Ratios

15.1 Cross Ratios

15.2 Applications of Cross Ratios

15.3 Problems

16 Introduction to Projective Geometry

16.1 Straightedge Constructions

16.2 Perspectivities and Projectivities

16.3 Line Perspectivities and Line Projectivities

16.4 Projective Geometry and Fixed Points

16.5 Projecting a Line to Infinity

16.6 The Apollonian Definition of a Conic

16.7 Problems

Bibliography

Index

The users who browse this book also browse


No browse record.